If A = L1D1U1 and A = L2D2U2, prove that L1 = L2, D1 = D2, and U1 = U2. If A is

For the equations x + y = 4, 2x 2y = 4, draw the row picture (two intersecting lines) and the column picture (combination of two columns equal to the column vector (4,4) on the right side).

Solve to find a combination of the columns that equals b: Triangular system u v w = b1 v + w = b2 w = b3

(Recommended) Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u+w = 2 (all in four-dimensional space). Is it a line or a point or an empty set? What is the intersection if the fourth plane u = 1 is included? Find a fourth equation that leaves us with no solution.

Sketch these three lines and decide if the equations are solvable: 3 by 2 system x + 2y = 2 x y = 2 y = 1. What happens if all right-hand sides are zero? Is there any nonzero choice of righthand sides that allows the three lines to intersect at the same point?

Find two points on the line of intersection of the three planes t = 0 and z = 0 and x+y+z+t = 1 in four-dimensional space.

When b = (2,5,7), find a solution (u,v,w) to equation (4) different from the solution (1,0,1) mentioned in the text.

Give two more right-hand sides in addition to b = (2,5,7) for which equation (4) can be solved. Give two more right-hand sides in addition to b = (2,5,6) for which it cannot be solved.

Explain why the system u + v + w = 2 u + 2v + 3w = 1 v + 2w = 0 is singular by finding a combination of the three equations that adds up to 0 = 1. What value should replace the last zero on the right side to allow the equations to have solutionsand what is one of the solutions?

The column picture for the previous exercise (singular system) is u 1 1 0 +v 1 2 1 +w 1 3 2 = b. Show that the three columns on the left lie in the same plane by expressing the third column as a combination of the first two. What are all the solutions (u,v,w) if b is the zero vector (0,0,0)?

Recommended) Under what condition on y1, y2, y3 do the points (0,y1), (1,y2), (2,y3) lie on a straight line?

These equations are certain to have the solution x = y = 0. For which values of a is there a whole line of solutions? ax + 2y = 0 2x + ay = 0

Starting with x+4y = 7, find the equation for the parallel line through x = 0, y = 0. Find the equation of another line that meets the first at x = 3, y = 1.

Draw the two pictures in two planes for the equations x2y = 0, x+y = 6

For two linear equations in three unknowns x, y, z, the row picture will show (2 or 3) (lines or planes) in (two or three)-dimensional space. The column picture is in (two or three)-dimensional space. The solutions normally lie on a .

For four linear equations in two unknowns x and y, the row picture shows four . The column picture is in -dimensional space. The equations have no solution unless the vector on the right-hand side is a combination of .

Find a point with z = 2 on the intersection line of the planes x + y + 3z = 6 and xy+z = 4. Find the point with z = 0 and a third point halfway between

The first of these equations plus the second equals the third: x + y + z = 2 x + 2y + z = 3 2x + 3y + 2z = 5. The first two planes meet along a line. The third plane contains that line, because if x, y, z satisfy the first two equations then they also . The equations have infinitely many solutions (the whole line L). Find three solutions.

Move the third plane in Problem 17 to a parallel plane 2x + 3y + 2z = 9. Now the three equations have no solutionwhy not? The first two planes meet along the line L, but the third plane doesnt that line.

In Problem 17 the columns are (1,1,2) and (1,2,3) and (1,1,2). This is a singular case because the third column is . Find two combinations of the columns that give b = (2,3,5). This is only possible for b = (4,6,c) if c = .

Normally 4 planes in four-dimensional space meet at a . Normally 4 column vectors in four-dimensional space can combine to produce b. What combination of (1,0,0,0), (1,1,0,0), (1,1,1,0), (1,1,1,1) produces b = (3,3,3,2)? What 4 equations for x, y, z, t are you solving?

When equation 1 is added to equation 2, which of these are changed: the planes in the row picture, the column picture, the coefficient matrix, the solution?

If (a,b) is a multiple of (c,d) with abcd 6= 0, show that (a,c) is a multiple of (b,d). This is surprisingly important: call it a challenge question. You could use numbers first to see how a, b, c, and d are related. The question will lead to: If A = a b c d has dependent rows then it has dependent columns.

. In these equations, the third column (multiplying w) is the same as the right side b. The column form of the equations immediately gives what solution for (u,v,w)? 6u + 7v + 8w = 8 4u + 5v + 9w = 9 2u 2v + 7w = 7.

What multiple ` of equation 1 should be subtracted from equation 2? 2x + 3y = 1 10x + 9y = 11. After this elimination step, write down the upper triangular system and circle the two pivots. The numbers 1 and 11 have no influence on those pivots.

Solve the triangular system of Problem 1 by back-substitution, y before x. Verify that x times (2,10) plus y times (3,9) equals (1,11). If the right-hand side changes to (4,44), what is the new solution?

What multiple of equation 2 should be subtracted from equation 3? 2x 4y = 6 x + 5y = 0. After this elimination step, solve the triangular system. If the right-hand side changes to (6,0), what is the new solution?

What multiple ` of equation 1 should be subtracted from equation 2? ax + by = f cx + dy = g. The first pivot is a (assumed nonzero). Elimination produces what formula for the second pivot? What is y? The second pivot is missing when ad = bc.

Choose a right-hand side which gives no solution and another right-hand side which gives infinitely many solutions. What are two of those solutions? 3x + 2y = 10 6x + 4y = .

Choose a coefficient b that makes this system singular. Then choose a right-hand side g that makes it solvable. Find two solutions in that singular case. 2x + by = 16 4x + 8y = g.

For which numbers a does elimination break down (a) permanently, and (b) temporarily? ax + 3y = 3 4x + 6y = 6. Solve for x and y after fixing the second breakdown by a row exchange.

For which three numbers k does elimination break down? Which is fixed by a row exchange? In each case, is the number of solutions 0 or 1 or ? kx + 3y = 6 3x + ky = 6.

What test on b1 and b2 decides whether these two equations allow a solution? How many solutions will they have? Draw the column picture. 3x 2y = b1 6x 4y = b2.

Reduce this system to upper triangular form by two row operations: 2x + 3y + z = 8 4x + 7y + 5z = 20 2y + 2z = 0. Circle the pivots. Solve by back-substitution for z, y, x.

Apply elimination (circle the pivots) and back-substitution to solve 2x 3y = 3 4x 5y + z = 7 2x y 3z = 5. List the three row operations: Subtract times row from row .

Which number d forces a row exchange, and what is the triangular system (not singular) for that d? Which d makes this system singular (no third pivot)? 2x + 5y + z = 0 4x + dy + z = 2 y z = 3.

Which number b leads later to a row exchange? Which b leads to a missing pivot? In that singular case find a nonzero solution x, y, z. x + by = 0 x 2y z = 0 y + z = 0.

(a) Construct a 3 by 3 system that needs two row exchanges to reach a triangular form and a solution. (b) Construct a 3 by 3 system that needs a row exchange to keep going, but breaks down later.

If rows 1 and 2 are the same, how far can you get with elimination (allowing row exchange)? If columns 1 and 2 are the same, which pivot is missing? 2xy+z = 0 2xy+z = 0 4x+y+z = 2 2x+2y+z = 0 4x+4y+z = 0 6x+6y+z = 2.

Construct a 3 by 3 example that has 9 different coefficients on the left-hand side, but rows 2 and 3 become zero in elimination. How many solutions to your system with b = (1,10,100) and how many with b = (0,0,0)?

Which number q makes this system singular and which right-hand side t gives it infinitely many solutions? Find the solution that has z = 1. x + 4y 2z = 1 x + 7y 6z = 6 3y + qz = t.

(Recommended) It is impossible for a system of linear equations to have exactly two solutions. Explain why. (a) If (x,y,z) and (X,Y,Z) are two solutions, what is another one? (b) If 25 planes meet at two points, where else do they meet?

Three planes can fail to have an intersection point, when no two planes are parallel. The system is singular if row 3 of A is a of the first two rows. Find a third equation that cant be solved if x+y+z = 0 and x2yz = 1.

Find the pivots and the solution for these four equations: 2x + y = 0 x + 2y + z = 0 y + 2z + t = 0 z + 2t = 5.

If you extend Problem 20 following the 1, 2, 1 pattern or the 1, 2, 1 pattern, what is the fifth pivot? What is the nth pivot?

Apply elimination and back-substitution to solve 2u + 3v = 0 4u + 5v + w = 3 2u v 3w = 5. What are the pivots? List the three operations in which a multiple of one row is subtracted from another.

For the system u + v + w = 2 u + 3v + 3w = 0 u + 3v + 5w = 2, what is the triangular system after forward elimination, and what is the solution?

Solve the system and find the pivots when 2u v = 0 u + 2v w = 0 v + 2w z = 0 w + 2z = 5. You may carry the right-hand side as a fifth column (and omit writing u, v, w, z until the solution at the end).

Apply elimination to the system u + v + w = 2 3u + 3v w = 6 u v + w = 1. When a zero arises in the pivot position, exchange that equation for the one below it and proceed. What coefficient of v in the third equation, in place of the present 1, would make it impossible to proceedand force elimination to break down?

Solve by elimination the system of two equations x y = 0 3x + 6y = 18. Draw a graph representing each equation as a straight line in the x-y plane; the lines intersect at the solution. Also, add one more linethe graph of the new second equation which arises after elimination.

Find three values of a for which elimination breaks down, temporarily or permanently, in au + u = 1 4u + av = 2. Breakdown at the first step can be fixed by exchanging rowsbut not breakdown at the last step.

True or false: (a) If the third equation starts with a zero coefficient (it begins with 0u) then no multiple of equation 1 will be subtracted from equation 3. (b) If the third equation has zero as its second coefficient (it contains 0v) then no multiple of equation 2 will be subtracted from equation 3. (c) If the third equation contains 0u and 0v, then no multiple of equation 1 or equation 2 will be subtracted from equation 3.

(Very optional) Normally the multiplication of two complex numbers (a+ib)(c+id) = (acbd) +i(bc+ad) involves the four separate multiplications ac, bd, be, ad. Ignoring i, can you compute acbd and bc+ad with only three multiplications? (You may do additions, such as forming a+b before multiplying, without any penalty.)

Use elimination to solve u + v + w = 6 u + 2v + 2w = 11 2u + 3v 4w = 3 and u + v + w = 7 u + 2v + 2w = 10 2u + 3v 4w = 3.

For which three numbers a will elimination fail to give three pivots? ax+2y+3z = b1 ax+ay+4z = b2 ax+ay+az = b3.

Find experimentally the average size (absolute value) of the first and second and third pivots for MATLABs lu(rand(3,3)). The average of the first pivot from abs(A(1,1)) should be 0.5.

Compute the products 4 0 1 0 1 0 4 0 1 3 4 5 and 1 0 0 0 1 0 0 0 1 5 2 3 and " 2 0 1 3#"1 1 # . For the third one, draw the column vectors (2,1) and (0,3). Multiplying by (1,1) just adds the vectors (do it graphically).

Working a column at a time, compute the products 4 1 5 1 6 1 " 1 3 # and 1 2 3 4 5 6 7 8 9 0 1 0 and 4 3 6 6 8 9 " 1 2 1 3 #

Find two inner products and a matrix product: h 1 2 7i 1 2 7 and h 1 2 7i 3 5 1 and 1 2 7 h 3 5 1i . The first gives the length of the vector (squared).

If an m by n matrix A multiplies an n-dimensional vector x, how many separate multiplications are involved? What if A multiplies an n by p matrix B?

Multiply Ax to find a solution vector x to the system Ax = zero vector. Can you find more solutions to Ax = 0? Ax = 3 6 0 0 2 2 1 1 1 2 1 1

Write down the 2 by 2 matrices A and B that have entries ai j = i+ j and bi j = (1) i+j . Multiply them to find AB and BA.

Give 3 by 3 examples (not just the zero matrix) of (a) a diagonal matrix: ai j = 0 if i 6= j. (b) a symmetric matrix: ai j = aji for all i and j. (c) an upper triangular matrix: ai j = 0 if i > j. (d) a skew-symmetric matrix: ai j = aji for all i and j.

Do these subroutines multiply Ax by rows or columns? Start with B(I) = 0: DO 10 I = 1, N DO 10 J = 1, N DO 10 J = 1, N DO 10 I = 1, N 10 B(I) = B(I) + A(I,J) * X(J) 10 B(I) = B(I) + A(I,J) * X(J) The outputs Bx = Ax are the same. The second code is slightly more efficient in FORTRAN and much more efficient on a vector machine (the first changes single entries B(I), the second can update whole vectors).

If the entries of A are ai j, use subscript notation to write (a) the first pivot. (b) the multiplier `i1 of row 1 to be subtracted from row i. (c) the new entry that replaces ai j after that subtraction. (d) the second pivot.

True or false? Give a specific counterexample when false. (a) If columns 1 and 3 of B are the same, so are columns 1 and 3 of AB. (b) If rows 1 and 3 of B are the same, so are rows 1 and 3 of AB. (c) If rows 1 and 3 of A are the same, so are rows 1 and 3 of AB. (d) (AB) 2 = A 2B 2 .

The first row of AB is a linear combination of all the rows of B. What are the coeffi- cients in this combination, and what is the first row of AB, if A = " 2 1 4 0 1 1# and B = 1 1 0 1 1 0 ?

The product of two lower triangular matrices is again lower triangular (all its entries above the main diagonal are zero). Confirm this with a 3 by 3 example, and then explain how it follows from the laws of matrix multiplication.

By trial and error find examples of 2 by 2 matrices such that (a) A 2 = I, A having only real entries. (b) B 2 = 0, although B 6= 0. (c) CD = DC, not allowing the case CD = 0. (d) EF = 0, although no entries of E or F are zero.

Describe the rows of EA and the columns of AE if E = " 1 7 0 1#

Suppose A commutes with every 2 by 2 matrix (AB = BA), and in particular A = " a b c d# commutes with B1 = " 1 0 0 0# and B2 = " 0 1 0 0# . Show that a = d and b = c = 0. If AB = BA for all matrices B, then A is a multiple of the identity.

Let x be the column vector (1,0,...,0). Show that the rule (AB)x = A(Bx) forces the first column of AB to equal A times the first column of B.

Which of the following matrices are guaranteed to equal (A+B) 2 ? A 2 +2AB+B 2 , A(A+B) +B(A+B), (A+B)(B+A), A 2 +AB+BA+B 2 .

If A and B are n by n matrices with all entries equal to 1, find (AB)i j. Summation notation turns the product AB, and the law (AB)C = A(BC), into (AB)i j = k aikbk j j k aikbk j! c jl = k aik j bk jc jl! . Compute both sides if C is also n by n, with every c jl = 2.

A fourth way to multiply matrices is columns of A times rows of B: AB = (column 1)(row 1) ++ (column n)(row n) = sum of simple matrices. Give a 2 by 2 example of this important rule for matrix multiplication

The matrix that rotates the x-y plane by an angle is A() = " cos sin sin cos # . Verify that A(1)A(2) = A(1+2) from the identities for cos(1+2) and sin(1+ 2). What is A() times A()?

Find the powers A 2 , A 3 (A 2 times A), and B 2 , B 3 , C 2 , C 3 . What are A k , B k , and C k ? A = " 1 2 1 2 1 2 1 2 # and B = " 1 0 0 1 # and C = AB = " 1 2 1 2 1 2 1

Write down the 3 by 3 matrices that produce these elimination steps: (a) E21 subtracts 5 times row 1 from row 2. (b) E32 subtracts 7 times row 2 from row 3. (c) P exchanges rows 1 and 2, then rows 2 and 3.

. In Problem 22, applying E21 and then E32 to the column b = (1,0,0) gives E32E21b = . Applying E32 before E21 gives E21E32b = . When E32 comes first, row feels no effect from row .

Which three matrices E21, E31, E32 put A into triangular form U? A = 1 1 0 4 6 1 2 2 0 and E32E31E21A = U. Multiply those Es to get one matrix M that does elimination: MA = U.

Suppose a33 = 7 and the third pivot is 5. If you change a33 to 11, the third pivot is . If you change a33 to , there is zero in the pivot position.

If every column of A is a multiple of (1,1,1), then Ax is always a multiple of (1,1,1). Do a 3 by 3 example. How many pivots are produced by elimination?

What matrix E31 subtracts 7 times row 1 from row 3? To reverse that step, R31 should 7 times row to row . Multiply E31 by R31.

(a) E21 subtracts row 1 from row 2 and then P23 exchanges rows 2 and 3. What matrix M = P23E21 does both steps at once? (b) P23 exchanges rows 2 and 3 and then E31 subtracts row I from row 3. What matrix M = E31P23 does both steps at once? Explain why the Ms are the same but the Es are different.

(a) What 3 by 3 matrix E13 will add row 3 to row 1? (b) What matrix adds row 1 to row 3 and at the same time adds row 3 to row 1? (c) What matrix adds row 1 to row 3 and then adds row 3 to row 1?

Multiply these matrices: 0 0 1 0 1 0 1 0 0 1 2 3 4 5 6 7 8 9 0 0 1 0 1 0 1 0 0 and 1 0 0 1 1 0 1 0 1 1 2 3 1 3 1 1 4 0

This 4 by 4 matrix needs which elimination matrices E21 and E32 and E43? A = 2 1 0 0 1 2 1 0 0 1 2 1 0 0 1 2

Write these ancient problems in a 2 by 2 matrix form Ax = b and solve them: (a) X is twice as old as Y and their ages add to 39, (b) (x,y) = (2,5) and (3,7) lie on the line y = mx+c. Find m and c.

The parabola y = a+bx+cx2 goes through the points (x,y) = (1,4) and (2,8) and (3,14). Find and solve a matrix equation for the unknowns (a,b,c).

Multiply these matrices in the orders EF and FE and E 2 : E = 1 0 0 a 1 0 b 0 1 F = 1 0 0 0 1 0 0 c 1

(a) Suppose all columns of B are the same. Then all columns of EB are the same, because each one is E times . (b) Suppose all rows of B are [1 2 4]. Show by example that all rows of EB are not [1 2 4]. It is true that those rows are .

If E adds row 1 to row 2 and F adds row 2 to row 1, does EF equal FE?

The first component of Ax is a1 jx j = a11x1 + + a1nxn. Write formulas for the third component of Ax and the (1,1) entry of A 2 .

If AB = I and BC = I, use the associative law to prove A = C.

A is 3 by 5, B is 5 by 3, C is 5 by 1, and D is 3 by 1. All entries are 1. Which of these matrix operations are allowed, and what are the results? BA AB ABD DBA A(B+C).

What rows or columns or matrices do you multiply to find (a) the third column of AB? (b) the first row of AB? (c) the entry in row 3, column 4 of AB? (d) the entry in row 1, column 1 of CDE?

(3 by 3 matrices) Choose the only B so that for every matrix A, (a) BA = 4A. (b) BA = 4B. (c) BA has rows 1 and 3 of A reversed and row 2 unchanged. (d) All rows of BA are the same as row 1 of A.

True or false? (a) If A 2 is defined then A is necessarily square. (b) If AB and BA are defined then A and B are square. (c) If AB and BA are defined then AB and BA are square. (d) If AB = B then A = I.

If A is m by n, how many separate multiplications are involved when (a) A multiplies a vector x with n components? (b) A multiplies an n by p matrix B? Then AB is m by p. (c) A multiplies itself to produce A 2 ? Here m = n.

To prove that (AB)C = A(BC), use the column vectors b1,...,bn of B. First suppose that C has only one column c with entries c1,...,cn: AB has columns Ab1,...,Abn, and Bc has one column c1b1 ++cnbn. Then (AB)c = c1Ab1 ++cnAbn, equals A(c1b1 ++cnbn) = A(Bc). Linearity gives equality of those two sums, and (AB)c = A(Bc). The same is true for all other of C. Therefore (AB)C = A(BC).

Multiply AB using columns times rows: AB = 1 0 2 4 2 1 " 3 3 0 1 2 1# = 1 2 2 h 3 3 0i + = .

Block multiplication separates matrices into blocks (submatrices). If their shapes make block multiplication possible, then it is allowed. Replace these xs by numbers and confirm that block multiplication succeeds. h A Bi " C D # = h AC +BDi and x x x x x x x x x x x x x x x x x x

Draw the cuts in A and B and AB to show how each of the four multiplication rules is really a block multiplication to find AB: (a) Matrix A times columns of B. (b) Rows of A times matrix B. (c) Rows of A times columns of B. (d) Columns of A times rows of B

Block multiplication says that elimination on column 1 produces EA = " 1 0 c/a I#"a b c D # = " a b 0 # .

Elimination for a 2 by 2 block matrix: When A 1A = I, multiply the first block row by CA1 and subtract from the second row, to find the Schur complement S: " I 0 CA1 I #"A B C D# = " A B 0 S #

With i 2 = 1, the product (A + iB)(x + iy) is Ax + iBx + iAy By. Use blocks to separate the real part from the imaginary part that multiplies i: " A B ? ? #"x y # = " AxBy ? # real part imaginary part

Suppose you solve Ax = b for three special right-hand sides b: Ax1 = 1 0 0 and Ax2 = 0 1 0 and Ax3 = 0 0 1 . If the solutions x1, x2, x3 are the columns of a matrix X, what is AX?

If the three solutions in Question 51 are x1 = (1,1,1) and x2 = (0,1,1) and x3 = (0,0,1), solve Ax = b when b = (3,5,8). Challenge problem: What is A?

Find all matrices A = " a b c d# that satisfy A " 1 1 1 1# = " 1 1 1 1# A.

If you multiply a northwest matrix A and a southeast matrix B, what type of matrices are AB and BA? Northwest and southeast mean zeros below and above the antidiagonal going from (1,n) to (n,1).

Write 2x+3y+z+5t = 8 as a matrix A (how many rows?) multiplying the column vector (x,y,z,t) to produce b. The solutions fill a plane in four-dimensional space. The plane is three-dimensional with no 4D volume

What 2 by 2 matrix P1 projects the vector (x,y) onto the x axis to produce (x,0)? What matrix P2 projects onto the y axis to produce (0,y)? If you multiply (5,7) by P1 and then multiply by P2, you get ( ) and ( ).

Write the inner product of (1,4,5) and (x,y,z) as a matrix multiplication Ax. A has one row. The solutions to Ax = 0 lie on a perpendicular to the vector . The columns of A are only in -dimensional space.

In MATLAB notation, write the commands that define the matrix A and the column vectors x and b. What command would test whether or not Ax = b? A = " 1 2 3 4# x = " 5 2 # b = " 1 7 #

The MATLAB commands A = eye(3) and v = [3:5] produce the 3 by 3 identity matrix and the column vector (3,4,5). What are the outputs from A v and v v? (Computer not needed!) If you ask for v A, what happens?

If you multiply the 4 by 4 all-ones matrix A = ones(4,4) and the column v = ones(4,1), what is A v? (Computer not needed.) If you multiply B = eye(4) + ones(4,4) times w = zeros(4,1) + 2 ones(4,1), what is B w?

Invent a 3 by 3 magic matrix M with entries 1,2,...,9. All rows and columns and diagonals add to 15. The first row could be 8, 3, 4. What is M times (1,1,1)? What is the row vector h 1 1 1i times M?

When is an upper triangular matrix nonsingular (a full set of pivots)?

What multiple `32 of row 2 of A will elimination subtract from row 3 of A? Use the factored form A = 1 0 0 2 1 0 1 4 1 5 7 8 0 2 3 0 0 6 . What will be the pivots? Will a row exchange be required?

Multiply the matrix L = E 1F 1G 1 in equation (6) by GFE in equation (3): 1 0 0 2 1 0 1 1 1 times 1 0 0 2 1 0 1 1 1 . Multiply also in the opposite order. Why are the answers what they are?

Apply elimination to produce the factors L and U for A = " 2 1 8 7# and A = 3 1 1 1 3 1 1 1 3 and A = 1 1 1 1 4 4 1 4 8

Factor A into LU, and write down the upper triangular system Ux = c which appears after elimination, for Ax = 2 3 3 0 5 7 6 9 8 u v w = 2 2 5

Find E 2 and E 8 and E 1 if E = " 1 0 6 1#

Find the products FGH and HGF if (with upper triangular zeros omitted) F = 1 2 1 0 0 1 0 0 0 1 G = 1 0 1 0 2 1 0 0 0 1 H = 1 0 1 0 0 1 0 0 2 1

(Second proof of A = LU) The third row of U comes from the third row of A by subtracting multiples of rows 1 and 2 (of U!): row 3 of U = row 3 of A`31(row 1 of U)`32(row 2 of U). (a) Why are rows of U subtracted off and not rows of A? Answer: Because by the time a pivot row is used, . (b) The equation above is the same as row 3 of A = `31(row 1 of U) +`32(row 2 of U) +1(row 3 of U). Which rule for matrix multiplication makes this row 3 of L times U? The other rows of LU agree similarly with the rows of A.

(a) Under what conditions is the following product nonsingular? A = 1 0 0 1 1 0 0 1 1 d1 d2 d3 1 1 0 0 1 1 0 0 1 . (b) Solve the system Ax = b starting with Lc = b: 1 0 0 1 1 0 0 1 1 c1 c2 c3 = 0 0 1 = b.

(a) Why does it take approximately n 2/2 multiplication-subtraction steps to solve each of Lc = b and Ux = c? (b) How many steps does elimination use in solving 10 systems with the same 60 by 60 coefficient matrix A?

Solve as two triangular systems, without multiplying LU to find A: LUx = 1 0 0 1 1 0 1 0 1 2 4 4 0 1 2 0 0 1 u v w = 2 0 2

How could you factor A into a product UL, upper triangular times lower triangular? Would they be the same factors as in A = LU?

Solve by elimination, exchanging rows when necessary: u + 4v + 2w = 2 2u 8v + 3w = 32 v + w = 1 and v + w = 0 u + v = 0 u + v + w = 1. Which permutation matrices are required?

Write down all six of the 3 by 3 permutation matrices, including P = I. Identify their inverses, which are also permutation matrices. The inverses satisfy PP1 = I and are on the same list.

Find the PA = LDU factorizations (and check them) for A = 0 1 1 1 0 1 2 3 4 and A = 1 2 1 2 4 2 1 1 1

Find a 4 by 4 permutation matrix that requires three row exchanges to reach the end of elimination (which is U = I).

The less familiar form A = LPU exchanges rows only at the end: A = 1 1 1 1 1 3 2 5 8 L 1A = 1 1 1 0 0 2 0 3 6 = PU = 1 0 0 0 0 1 0 1 0 1 1 1 0 3 6 0 0 2 . What is L is this case? Comparing with PA = LU in Box 1J, the multipliers now stay in place (`21 is 1 and `31 is 2 when A = LPU).

Decide whether the following systems are singular or nonsingular, and whether they have no solution, one solution, or infinitely many solutions: v w = 2 u v = 2 u w = 2 and v w = 0 u v = 0 u w = 0 and v + w = 1 u + v = 1 u + w = 1.

Which numbers a, b, c lead to row exchanges? Which make the matrix singular? A = 1 2 0 a 8 3 0 b 5 and A = " c 2 6 4# .

Forward elimination changes 1 1 1 2 x = b to a triangular 1 1 0 1 x = c: x + y = 5 x + 2y = 7 x + y = 5 y = 2 " 1 1 5 1 2 7 # " 1 1 5 0 1 2 # . That step subtracted `21 = times row 1 from row 2. The reverse step adds `21 times row 1 to row 2. The matrix for that reverse step is L = . Multiply this L times the triangular system 1 1 0 1 x = 5 2 to get = . In letters, L multiplies Ux = c to give .

(Move to 3 by 3) Forward elimination changes Ax = b to a triangular Ux = c: x+y+z = 5 x+2y+3z = 7 x+3y+6z = 11 x+y+z = 5 y+2z = 2 2y+5z = 6 x+y+z = 5 y+2z = 2 z = 2. The equation z = 2 in Ux = c comes from the original x+3y+6z = 11 in Ax = b by subtracting `31 = times equation 1 and `32 = times the final equation 2. Reverse that to recover [1 3 6 11] in [A b] from the final [1 1 1 5] and [0 1 2 2] and [0 0 1 2] in [U c]: Row 3 of h A bi = (`31 Row 1 +`32 Row 2+1 Row 3) of h U ci . In matrix notation this is multiplication by L. So A = LU and b = Lc.

What are the 3 by 3 triangular systems Lc = b and Ux = c from Problem 21? Check that c = (5,2,2) solves the first one. Which x solves the second one?

What two elimination matrices E21 and E32 put A into upper triangular form E32E21A = U? Multiply by E 1 31 and E 1 21 to factor A into LU = E 1 21 E 1 32 U: A = 1 1 1 2 4 5 0 4 0

What three elimination matrices E21, E31, E32 put A into upper triangular form E32E31E21A = U? Multiply by E 1 32 , E 1 31 and E 1 21 to factor A into LU where L = E 1 21 E 1 31 E 1 32 . Find L and U: A = 1 0 1 2 2 2 3 4 5

When zero appears in a pivot position, A = LU is not possible! (We need nonzero pivots d, f , i in U.) Show directly why these are both impossible: " 0 1 2 3# = " 1 0 ` 1 #"d e 0 f # 1 1 0 1 1 2 1 2 1 = 1 ` 1 m n 1 d e g f h i

Which number c leads to zero in the second pivot position? A row exchange is needed and A = LU is not possible. Which c produces zero in the third pivot position? Then a row exchange cant help and elimination fails: A = 1 c 0 2 4 1 3 5 1

What are L and D for this matrix A? What is U in A = LU and what is the new U in A = LDU? A = 2 4 8 0 3 9 0 0 7

A and B are symmetric across the diagonal (because 4 = 4). Find their triple factorizations LDU and say how U is related to L for these symmetric matrices: A = " 2 4 4 11# and B = 1 4 0 4 12 4 0 4 0

(Recommended) Compute L and U for the symmetric matrix A = a a a a a b b b a b c c a b c d . Find four conditions on a, b, c, d to get A = LU with four pivots.

Find L and U for the nonsymmetric matrix A = a r r r a b s s a b c t a b c d . Find the four conditions on a, b, c, d, r, s, t to get A = LU with four pivots.

Tridiagonal matrices have zero entries except on the main diagonal and the two adjacent diagonals. Factor these into A = LU and A = LDV: A = 1 1 0 1 2 1 0 1 2 and A = a a 0 a a+b b 0 b b+c

Solve the triangular system Lc = b to find c. Then solve Ux = c to find x: L = " 1 0 4 1# and U = " 2 4 0 1# and b = " 2 11# . For safety find A = LU and solve Ax = b as usual. Circle c when you see it.

Solve Lc = b to find c. Then solve Ux = c to find x. What was A? L = 1 0 0 1 1 0 1 1 1 and U = 1 1 1 0 1 1 0 0 1 and b = 4 5 6

If A and B have nonzeros in the positions marked by x, which zeros are still zero in their factors L and U? A = x x x x x x x 0 0 x x x 0 0 x x and B = x x x 0 x x 0 x x 0 x x 0 x x x

(Important) If A has pivots 2, 7, 6 with no row exchanges, what are the pivots for the upper left 2 by 2 submatrix B (without row 3 and column 3)? Explain why.

Starting from a 3 by 3 matrix A with pivots 2, 7, 6, add a fourth row and column to produce M. What are the first three pivots for M, and why? What fourth row and column are sure to produce 9 as the fourth pivot?

Use chol(pascal(5)) to find the triangular factors of MATLABs pascal(5). Row exchanges in [L, U] = lu(pascal(5)) spoil Pascals pattern

(Review) For which numbers c is A = LU impossiblewith three pivots? A = 1 2 0 3 c 1 0 1 1

Estimate the time difference for each new right-hand side b when n = 800. Create A = rand(800) and b = rand(800,1) and B = rand(800,9). Compare the times from tic; A\b; toc and tic; A\B; toc (which solves for 9 right sides).

There are 12 even permutations of (1,2,3,4), with an even number of exchanges. Two of them are (1,2,3,4) with no exchanges and (4,3,2,1) with two exchanges. List the other ten. Instead of writing each 4 by 4 matrix, use the numbers 4, 3, 2, 1 to give the position of the 1 in each row.

How many exchanges will permute (5,4,3,2,1) back to (1,2,3,4,5)? How many exchanges to change (6,5,4,3,2,1) to (1,2,3,4,5,6)? One is even and the other is odd. For (n,...,1) to (1,...,n), show that n = 100 and 101 are even, n = 102 and 103 are odd.

If P1 and P2 are permutation matrices, so is P1P2. This still has the rows of I in some order. Give examples with P1P2 6= P2P1 and P3P4 = P4P3.

(Try this question.) Which permutation makes PA upper triangular? Which permutations make P1AP2 lower triangular? Multiplying A on the right by P2 exchanges the of A. A = 0 0 6 1 2 3 0 4 5

Find a 3 by 3 permutation matrix with P 3 = I (but not P = I). Find a 4 by 4 permutation Pb with Pb4 6= I.

If you take powers of a permutation, why is some P k eventually equal to I? Find a 5 by 5 permutation P so that the smallest power to equal I is P 6 . (This is a challenge question. Combine a 2 by 2 block with a 3 by 3 block.)

The matrix P that multiplies (x,y,z) to give (z,x,y) is also a rotation matrix. Find P and P 3 . The rotation axis a = (1,1,1) doesnt move, it equals Pa. What is the angle of rotation from v = (2,3,5) to Pv = (5,2,3)?

If P is any permutation matrix, find a nonzero vector x so that (I P)x = 0. (This will mean that I P has no inverse, and has determinant zero.)

If P has 1s on the antidiagonal from (1,n) to (n,1), describe PAP.

Find the inverses (no special system required) of A1 = " 0 2 3 0# , A2 = " 2 0 4 2# , A3 = " cos sin sin cos

(a) Find the inverses of the permutation matrices P = 0 0 1 0 1 0 1 0 0 and P = 0 0 1 1 0 0 0 1 0 . (b) Explain for permutations why P 1 is always the same as PT . Show that the 1s are in the right places to give PPT = I.

From AB = C find a formula for A 1 . Also find A 1 from PA = LU.

(a) If A is invertible and AB = AC, prove quickly that B = C. (b) If A = [ 1 0 0 0 ], find an example with AB = AC but B 6= C

(a) If A is invertible and AB = AC, prove quickly that B = C. (b) If A = [ 1 0 0 0 ], find an example with AB = AC but B 6= C

Use the Gauss-Jordan method to invert A1 = 1 0 0 1 1 1 0 0 1 , A2 = 2 1 0 1 2 1 0 1 2 , A3 = 0 0 1 0 1 1 1 1 1

. Find three 2 by 2 matrices, other than A = I and A = I, that are their own inverses: A 2 = I.

Show that A = [ 1 1 3 3 ] has no inverse by solving Ax = 0, and by failing to solve " 1 1 3 3#"a b c d# = " 1 0 0 1# .

Suppose elimination fails because there is no pivot in column 3: Missing pivot A = 2 1 4 6 0 3 8 5 0 0 0 7 0 0 0 9 Show that A cannot be invertible. The third row of A1, multiplying A, should give the third row [0 0 1 0] of A 1A = I. Why is this impossible?

Find the inverses (in any legal way) of A1 = 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 0 , A2 = 1 0 0 0 1 2 1 0 0 0 2 3 1 0 0 0 3 4 1 , A3 = a b 0 0 c d 0 0 0 0 a b 0 0 c d

Give examples of A and B such that (a) A+B is not invertible although A and B are invertible. (b) A+B is invertible although A and B are not invertible. (c) all of A, B, and A+B are invertible. (d) In the last case use A 1 (A+B)B 1 = B 1 +A 1 to show that C = B 1 +A 1 is also invertibleand find a formula for C 1

If A is invertible, which properties of A remain true for A 1 ? (a) A is triangular. (b) A is symmetric. (c) A is tridiagonal. (d) All entries are whole numbers. (e) All entries are fractions (including numbers like 3 1 ).

If A is invertible, which properties of A remain true for A 1 ? (a) A is triangular. (b) A is symmetric. (c) A is tridiagonal. (d) All entries are whole numbers. (e) All entries are fractions (including numbers like 3 1 ).

If B is square, show that A = B+B T is always symmetric and K = BB T is always skew-symmetricwhich means that K T = K. Find these matrices A and K when B = [ 1 3 1 1 ], and write B as the sum of a symmetric matrix and a skew-symmetric matrix.

(a) How many entries can be chosen independently in a symmetric matrix of order n? (b) How many entries can be chosen independently in a skew-symmetric matrix (K T = K) of order n? The diagonal of K is zero!

(a) If A = LDU, with 1s on the diagonals of L and U, what is the corresponding factorization of A T ? Note that A and A T (square matrices with no row exchanges) share the same pivots. (b) What triangular systems will give the solution to A T y = b?

If A = L1D1U1 and A = L2D2U2, prove that L1 = L2, D1 = D2, and U1 = U2. If A is invertible, the factorization is unique. (a) Derive the equation L 1 1 L2D2 = D1U1U 1 2 , and explain why one side is lower triangular and the other side is upper triangular. (b) Compare the main diagonals and then compare the off-diagonals.

Under what conditions on their entries are A and B invertible? A = a b c d e 0 f 0 0 B = a b 0 c d 0 0 0 e

Compute the symmetric LDLT factorization of A = 1 3 5 3 12 18 5 18 30 and A = " a b b d#

Find the inverse of A = 1 0 0 0 1 4 1 0 0 1 3 1 3 1 0 1 2 1 2 1 2 1

(Remarkable) If A and B are square matrices, show that I BA is invertible if I AB is invertible. Start from B(I AB) = (1BA)B.

Find the inverses (directly or from the 2 by 2 formula) of A, B, C: A = " 0 3 4 6# and B = " a b b 0 # and C = " 3 4 5 7

Solve for the columns and

Show that has no inverse by trying to solve for the column : must include

(Important) If A has row 1 + row 2 = row 3, show that A is not invertible: (a) Explain why Ax = (1,0,0) cannot have a solution. (b) Which right-hand sides (b1,b2,b3) might allow a solution to Ax = b? (c) What happens to row 3 in elimination?

If A has column 1 + column 2 = column 3, show that A is not invertible: (a) Find a nonzero solution x to Ax = 0. The matrix is 3 by 3. (b) Elimination keeps column 1 + column 2 = column 3. Explain why there is no third pivot.

Suppose A is invertible and you exchange its first two rows to reach B. Is the new matrix B invertible? How would you find B 1 from A 1 ?

If the product M = ABC of three square matrices is invertible, then A, B, C are invertible. Find a formula for B 1 that involves M1 and A and C.

Prove that a matrix with a column of zeros cannot have an inverse.

Multiply [ a b c d ] times [ d b c a ]. What is the inverse of each matrix if ad 6= bc?

(a) What matrix E has the same effect as these three steps? Subtract row 1 from row 2, subtract row 1 from row 3, then subtract row 2 from row 3. (b) What single matrix L has the same effect as these three reverse steps? Add row 2 to row 3, add row 1 to row 3, then add row 1 to row 2.

Find the numbers a and b that give the inverse of 5 eye(4) ones(4,4): 4 1 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 = a b b b b a b b b b a b b b b a . What are a and b in the inverse of 6 eye(5) ones(5,5)?

Show that A = 4 eye(4) ones(4,4) is not invertible: Multiply A ones(4,1).

There are sixteen 2 by 2 matrices whose entries are 1s and 0s. How many of them are invertible?

Change I into A 1 as you reduce A to I (by row operations): h A Ii = " 1 3 1 0 2 7 0 1# and h A Ii = " 1 4 1 0 3 9 0 1

Follow the 3 by 3 text example but with plus signs in A. Eliminate above and below the pivots to reduce [A I] to [I A1 ]: h A I i = 2 1 0 1 0 0 1 2 1 0 1 0 0 1 2 0 0 1

Use Gauss-Jordan elimination on [A I] to solve AA1 = I: 1 a b 0 1 c 0 0 1 h x1 x2 x3 i = 1 0 0 0 1 0 0 0 1

Invert these matrices A by the Gauss-Jordan method starting with [A I]: A = 1 0 0 2 1 3 0 0 1 and A = 1 1 1 1 2 2 1 2 3

Exchange rows and continue with Gauss-Jordan to find A 1 : h A Ii = " 0 2 1 0 2 2 0 1

True or false (with a counterexample if false and a reason if true): (a) A 4 by 4 matrix with a row of zeros is not invertible. (b) A matrix with Is down the main diagonal is invertible. (c) If A is invertible then A 1 is invertible. (d) If A T is invertible then A is invertible.

For which three numbers c is this matrix not invertible, and why not? A = 2 c c c c c 8 7 c

Prove that A is invertible if a 6= 0 and a 6= b (find the pivots and A 1 ): A = a b b a a b a a a

This matrix has a remarkable inverse. Find A 1 by elimination on [A I]. Extend to a 5 by 5 alternating matrix and guess its inverse: A = 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1

If B has the columns of A in reverse order, solve (AB)x = 0 to show that AB is not invertible. An example will lead you to x.

Find and check the inverses (assuming they exist) of these block matrices: " I 0 C I# "A 0 C D# "0 I I D#

Use inv(S) to invert MATLABs 4 by 4 symmetric matrix S = pascal(4). Create Pascals lower triangular A = abs(pascal(4,1)) and test inv(S) = inv(A) inv(A).

If A = ones(4,4) and b = rand(4,1), how does MATLAB tell you that Ax = b has no solution? If b = ones(4,1), which solution to Ax = b is found by A\b?

M1 shows the change in A 1 (useful to know) when a matrix is subtracted from A. Check part 3 by carefully multiplying MM1 to get I: 1. M = I uvT and M1 = I +uvT/(1v Tu). 2. M = AuvT and M1 = A 1 +A 1uvTA 1/(1v TA 1u). 3. M = I UV and M1 = In +U(Im VU) 1V. 4. M = AUW1V and M1 = A 1 +A 1U(W VA1U) 1VA1 . The four identities come from the 1, 1 block when inverting these matrices: " I u v T 1 # " A u v T 1 # "In U V Im # "A U V W#

Find A T and A 1 and (A 1 ) T and (A T ) 1 for A = " 1 0 9 3# and also A = " 1 c c 0 #

Verify that (AB) T equals B TA T but those are different from A TB T : A = " 1 0 2 1# B = " 1 3 0 1# AB = " 1 3 2 7# . In case AB = BA (not generally true!), how do you prove that B TA T = A TB T ?

(a) The matrix (AB) 1 T comes from (A 1 ) T and (B 1 ) T . In what order? (b) If U is upper triangular then (U 1 ) T is triangular.

Show that A 2 = 0 is possible but A TA = 0 is not possible (unless A = zero matrix).

(a) The row vector x T times A times the column y produces what number? x TAy = h 0 1i " 1 2 3 4 5 6# 0 1 0 = . (b) This is the row x TA = times the column y = (0,1,0). (c) This is the row x T = [0 1] times the column Ay = .

When you transpose a block matrix M = [ A B C D ] the result is MT = . Test it. Under what conditions on A, B, C, D is the block matrix symmetric?

Explain why the inner product of x and y equals the inner product of Px and Py. Then (Px) T (Py) = x T y says that P TP = I for any permutation. With x = (1,2,3) and y = (1,4,2), choose P to show that (Px) T y is not always equal to x T (P T y).

If A = A T and B = B T , which of these matrices are certainly symmetric? (a) A 2 B 2 (b) (A+B)(AB) (c) ABA (d) ABAB.

If A = A T needs a row exchange, then it also needs a column exchange to stay symmetric. In matrix language, PA loses the symmetry of A but recovers the symmetry.

(a) How many entries of A can be chosen independently, if A = A T is 5 by 5? (b) How do L and D (5 by 5) give the same number of choices in LDLT ?

Suppose R is rectangular (m by n) and A is symmetric (m by m). (a) Transpose R TAR to show its symmetry. What shape is this matrix? (b) Show why R TR has no negative numbers on its diagonal.

Factor these symmetric matrices into A = LDLT . The matrix D is diagonal: A = " 1 3 3 2# and A = " 1 b b c# and A = 2 1 0 1 2 1 0 1 2

Wires go between Boston, Chicago, and Seattle. Those cities are at voltages xB, xC, xS. With unit resistances between cities, the three currents are in y: y = Ax is yBC yCS yBS = 1 1 0 0 1 1 1 0 1 xB xC xS (a) Find the total currents AT y out of the three cities. (b) Verify that (Ax)T y agrees with x T (ATy)six terms in both.

Producing x1 trucks and x2 planes requires x1 + 50x2 tons of steel, 40x1 + 1000x2 pounds of rubber, and 2x1+50x2 months of labor. If the unit costs y1, y2, y3 are $700 per ton, $3 per pound, and $3000 per month, what are the values of one truck and one plane? Those are the components of A T y

Ax gives the amounts of steel, rubber, and labor to produce x in Problem 62. Find A. Then (Ax) T y is the of inputs while x T (A T y) is the value of .

Here is a new factorization of A into triangular times symmetric: Start from A = LDU. Then A equals L(U T ) 1 times U TDU. Why is L(U T ) 1 triangular? Its diagonal is all 1s. Why is U TDU symmetric?

A group of matrices includes AB and A 1 if it includes A and B. Products and inverses stay in the group. Which of these sets are groups? Lower triangularmatrices L with is on the diagonal, symmetric matrices S, positive matrices M, diagonal invertible matrices D, permutation matrices P. Invent two more matrix groups.

If every row of a 4 by 4 matrix contains the numbers 0, 1, 2, 3 in some order, can the matrix be symmetric? Can it be invertible?

Prove that no reordering of rows and reordering of columns can transpose a typical matrix

A square northwest matrix B is zero in the southeast corner, below the antidiagonal that connects (1,n) to (n,1). Will B T and B 2 be northwest matrices? Will B 1 be northwest or southeast? What is the shape of BC = northwest times southeast? You are allowed to combine permutations with the usual L and U (southwest and northeast).

Compare tic; inv(A); toc for A = rand(500) and A = rand(1000). The n 3 count says that computing time (measured by tic; toc) should multiply by 8 when n is doubled. Do you expect these random A to be invertible?

I = eye(1000); A = rand(1000); B = triu(A); produces a random triangular matrix B. Compare the times for inv(B) and B\I. Backslash is engineered to use the zeros in B, while inv uses the zeros in I when reducing [B I] by Gauss-Jordan. (Compare also with inv(A) and A\I for the full matrix A.)

Show that L 1 has entries j/i for i j (the 1, 2, 1 matrix has this L): L = 1 0 0 0 1 2 1 0 0 0 2 3 1 0 0 0 3 4 1 and L 1 = 1 0 0 0 1 2 1 0 0 1 3 2 3 1 0 1 4 2 4 3 4 1 Test this pattern for L = eye(5) diag(1:5)\diag(1:4,1) and inv(L).

Write out the LDU = LDLT factors of A in equation (6) when n = 4. Find the determinant as the product of the pivots in D.

Modify a11 in equation (6) from a11 = 2 to a11 = 1, and find the LDU factors of this new tridiagonal matrix.

Find the 5 by 5 matrix A0 (h = 1 6 ) that approximates d 2u dx2 = f(x), du dx (0) = du dx (1) = 0, replacing these boundary conditions by u0 = u1 and u6 = u5. Check that your A0 times the constant vector (C,C,C,C,C), yields zero; A0 is singular. Analogously, if u(x) is a solution of the continuous problem, then so is u(x) +C.

Write down the 3 by 3 finite-difference matrix equation (h = 1 4 ) for d 2u dx2 +u = x, u(0) = u(1) = 0.

With h = 1 4 and f(x) = 4 2 sin2x, the difference equation (5) is 2 1 0 1 2 1 0 1 2 u1 u2 u3 = 2 4 1 0 1 . Solve for u1, u2, u3 and find their error in comparison with the true solution u = sin2x at x = 1 4 , x = 1 2 , and x = 3 4 .

What 5 by 5 system replaces (6) if the boundary conditions are changed to u(0) = 1, u(1) = 0?

Compute H 1 in two ways for the 3 by 3 Hilbert matrix H = 1 1 2 1 3 1 2 1 3 1 4 1 3 1 4 1 5 , first by exact computation and second by rounding off each number to three figures. This matrix H is ill-conditioned and row exchanges dont help.

For the same matrix H, compare the right-hand sides of Hx = b when the solutions are x = (1,1,1) and x = (0,6,3.6).

Solve Hx = b = (1,0,...,0) for the 10 by 10 Hilbert matrix with hi j = 1/(i+ j1), using any computer code for linear equations. Then change an entry of b by .0001 and compare the solutions.

Compare the pivots in direct elimination to those with partial pivoting for A = " .001 0 1 1000# . (This is actually an example that needs rescaling before elimination.)

Explain why partial pivoting produces multipliers `i j in L that satisfy |`i j| 1. Can you construct a 3 by 3 example with all |ai j| 1 whose last pivot is 4? This is the worst possible, since each entry is at most doubled when |`i j| 1.

(a) Write down the 3 by 3 matrices with entries ai j = i j and bi j = i j . (b) Compute the products AB and BA and A 2.

For the matrices A = " 1 0 2 1# and B = " 1 2 0 1# , compute AB and BA and A 1 and B 1 and (AB) 1 .

Find examp1es of 2 by 2 matrices with a12 = 1 2 for which (a) A 2 = I. (b) A 1 = A T . (c) A 2 = A.

Solve by elimination and back-substitution: u + w = 4 u + v = 3 u + v + w = 6 and v + w = 0 u + w = 0 u + v = 6

Factor the preceding matrices into A = LU or PA = LU.

(a) There are sixteen 2 by 2 matrices whose entries are 1s and 0s. How many are invertible? (b) (Much harder!) If you put 1s and 0s at random into the entries of a 10 by 10 matrix, is it more likely to be invertible or singular?

There are sixteen 2 by 2 matrices whose entries are 1s and 1s. How many are invertible?

How are the rows of EA related to the rows of A in the following cases? E = 1 0 0 0 2 0 4 0 1 or E = " 1 1 1 0 0 0# or E = 0 0 1 0 1 0 1 0 0

Write down a 2 by 2 system with infinitely many solutions.

If E is 2 by 2 and it adds the first equation to the second, what are E 2 and E 8 and 8E?

True or false, with reason if true or counterexample if false: (1) If A is invertible and its rows are in reverse order in B, then B is invertible. (2) If A and B are symmetric then AB is symmetric. (3) If A and B are invertible then BA is invertible. (4) Every nonsingular matrix can be factored into the product A = LU of a lower triangular L and an upper triangular U.

Solve Ax = b by solving the triangular systems Lc = b and Ux = c: A = LU = 1 0 0 4 1 0 1 0 1 2 2 4 0 1 3 0 0 1 , b = 0 0 1 . What part of A 1 have you found, with this particular b?

If possible, find 3 by 3 matrices B such that (1) BA = 2A for every A. (2) BA = 2B for every A. (3) BA has the first and last rows of A reversed. (4) BA has the first and last columns of A reversed

Find the value for c in the following n by n inverse: if A = n 1 1 1 n 1 1 1 1 1 n then A 1 = 1 n+1 c 1 1 1 c 1 1 1 1 1 c

For which values of k does kx + y = 1 x + ky = 1 have no solution, one solution, or infinitely many solutions?

Find the symmetric factorization A = LDLT of A = 1 2 0 2 6 4 0 4 11 and A = " a b b c

Suppose A is the 4 by 4 identity matrix except for a vector v in column 2: A = 1 v1 0 0 0 v2 0 0 0 v3 1 0 0 v4 0 1 (a) Factor A into LU, assuming v2 6= 0. (b) Find A1 , which has the same form as A.

Solve by elimination, or show that there is no solution: u + v + w = 0 u + 2v + 3w = 0 3u + 5v + 7w = 1 and u + v + w = 0 u + u + 3w = 0 3u + 5v + 7w = 1.

Describe the rows of DA and the columns of AD if D = [ 2 0 0 5 ].

(a) If A is invertible what is the inverse of A T ? (b) If A is also symmetric what is the transpose of A 1 ? (c) Illustrate both formulas when A = [ 2 1 1 1 ].

By experiment with n = 2 and n = 3, find " 2 3 0 0#n , " 2 3 0 1#n , " 2 3 0 1#1

Starting with a first plane u+2vw = 6, find the equation for (a) the parallel plane through the origin. (b) a second plane that also contains the points (6,0,0) and (2,2,0). (c) a third plane that meets the first and second in the point (4,1,0).

What multiple of row 2 is subtracted from row 3 in forward elimination of A? A = 1 0 0 2 1 0 0 5 1 1 2 0 0 1 5 0 0 1 . How do you know (without multiplying those factors) that A is invertible, symmetric, and tridiagonal? What are its pivots?

(a) What vector x will make Ax = column 1 of A + 2(column 3), for a 3 by 3 matrix A? (b) Construct a matrix that has column 1 + 2(column 3) = 0. Check that A is singular (fewer than 3 pivots) and explain why that must be the case

True or false, with reason if true and counterexample if false: (1) If L1U1 = L2U2 (upper triangular Us with nonzero diagonal, lower triangular Ls with unit diagonal), then L1 = L2 and U1 = U2. The LU factorization is unique. (2) If A 2 +A = I then A 1 = A+I. (3) If all diagonal entries of A are zero, then A is singular.

By experiment or the Gauss-Jordan method compute 1 0 0 ` 1 0 m 0 1 n , 1 0 0 ` 1 0 m 0 1 1 , 1 0 0 ` 1 0 0 m 1 1

Write down the 2 by 2 matrices that (a) reverse the direction of every vector. (b) project every vector onto the x2 axis. (c) turn every vector counterclockwise through 90. (d) reflect every vector through the 45 line x1 = x2.