 1.1.7.1: Write out the LDU = LDLT factors of A in equation (6) when n = 4. F...
 1.1.1: (a) Write down the 3 by 3 matrices with entries ai j = i j and bi j...
 1.1.5.1: When is an upper triangular matrix nonsingular (a full set of pivots)?
 1.1.6.1: Find the inverses (no special system required) of A1 = " 0 2 3 0# ,...
 1.1.2.1: For the equations x + y = 4, 2x 2y = 4, draw the row picture (two i...
 1.1.3.1: What multiple ` of equation 1 should be subtracted from equation 2?...
 1.1.4.1: 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 ...
 1.1.7.2: Modify a11 in equation (6) from a11 = 2 to a11 = 1, and find the LD...
 1.1.2: For the matrices A = " 1 0 2 1# and B = " 1 2 0 1# , compute AB and...
 1.1.5.2: What multiple `32 of row 2 of A will elimination subtract from row ...
 1.1.6.2: (a) Find the inverses of the permutation matrices P = 0 0 1 0 1 0 1...
 1.1.2.2: Solve to find a combination of the columns that equals b: Triangula...
 1.1.3.2: Solve the triangular system of by backsubstitution, y before x. Ve...
 1.1.4.2: Working a column at a time, compute the products 4 1 5 1 6 1 " 1 3 ...
 1.1.7.3: Find the 5 by 5 matrix A0 (h = 1 6 ) that approximates d 2u dx2 = f...
 1.1.3: Find examp1es of 2 by 2 matrices with a12 = 1 2 for which (a) A 2 =...
 1.1.5.3: Multiply the matrix L = E 1F 1G 1 in equation (6) by GFE in equatio...
 1.1.6.3: From AB = C find a formula for A 1 . Also find A 1 from PA = LU.
 1.1.2.3: (Recommended) Describe the intersection of the three planes u+v+w+z...
 1.1.3.3: What multiple of equation 2 should be subtracted from equation 3? 2...
 1.1.4.3: Find two inner products and a matrix product: h 1 2 7i 1 2 7 and h ...
 1.1.7.4: Write down the 3 by 3 finitedifference matrix equation (h = 1 4 ) ...
 1.1.4: Solve by elimination and backsubstitution: u + w = 4 u + v = 3 u +...
 1.1.5.4: Apply elimination to produce the factors L and U for A = " 2 1 8 7#...
 1.1.6.4: (a) If A is invertible and AB = AC, prove quickly that B = C. (b) I...
 1.1.2.4: Sketch these three lines and decide if the equations are solvable: ...
 1.1.3.4: What multiple ` of equation 1 should be subtracted from equation 2?...
 1.1.4.4: If an m by n matrix A multiplies an ndimensional vector x, how man...
 1.1.7.5: With h = 1 4 and f(x) = 4 2 sin2x, the difference equation (5) is 2...
 1.1.5: Factor the preceding matrices into A = LU or PA = LU.
 1.1.5.5: Factor A into LU, and write down the upper triangular system Ux = c...
 1.1.6.5: (a) If A is invertible and AB = AC, prove quickly that B = C. (b) I...
 1.1.2.5: Find two points on the line of intersection of the three planes t =...
 1.1.3.5: Choose a righthand side which gives no solution and another right...
 1.1.4.5: Multiply Ax to find a solution vector x to the system Ax = zero vec...
 1.1.7.6: What 5 by 5 system replaces (6) if the boundary conditions are chan...
 1.1.6: (a) There are sixteen 2 by 2 matrices whose entries are 1s and 0s. ...
 1.1.5.6: Find E 2 and E 8 and E 1 if E = " 1 0 6 1#
 1.1.6.6: Use the GaussJordan method to invert A1 = 1 0 0 1 1 1 0 0 1 , A2 =...
 1.1.2.6: When b = (2,5,7), find a solution (u,v,w) to equation (4) different...
 1.1.3.6: Choose a coefficient b that makes this system singular. Then choose...
 1.1.4.6: Write down the 2 by 2 matrices A and B that have entries ai j = i+ ...
 1.1.7.7: Compute H 1 in two ways for the 3 by 3 Hilbert matrix H = 1 1 2 1 3...
 1.1.7: There are sixteen 2 by 2 matrices whose entries are 1s and 1s. How ...
 1.1.5.7: Find the products FGH and HGF if (with upper triangular zeros omitt...
 1.1.6.7: . Find three 2 by 2 matrices, other than A = I and A = I, that are ...
 1.1.2.7: Give two more righthand sides in addition to b = (2,5,7) for which...
 1.1.3.7: For which numbers a does elimination break down (a) permanently, an...
 1.1.4.7: Give 3 by 3 examples (not just the zero matrix) of (a) a diagonal m...
 1.1.7.8: For the same matrix H, compare the righthand sides of Hx = b when ...
 1.1.8: How are the rows of EA related to the rows of A in the following ca...
 1.1.5.8: (Second proof of A = LU) The third row of U comes from the third ro...
 1.1.6.8: Show that A = [ 1 1 3 3 ] has no inverse by solving Ax = 0, and by ...
 1.1.2.8: Explain why the system u + v + w = 2 u + 2v + 3w = 1 v + 2w = 0 is ...
 1.1.3.8: For which three numbers k does elimination break down? Which is fix...
 1.1.4.8: Do these subroutines multiply Ax by rows or columns? Start with B(I...
 1.1.7.9: Solve Hx = b = (1,0,...,0) for the 10 by 10 Hilbert matrix with hi ...
 1.1.9: Write down a 2 by 2 system with infinitely many solutions.
 1.1.5.9: (a) Under what conditions is the following product nonsingular? A =...
 1.1.6.9: Suppose elimination fails because there is no pivot in column 3: Mi...
 1.1.2.9: The column picture for the previous exercise (singular system) is u...
 1.1.3.9: What test on b1 and b2 decides whether these two equations allow a ...
 1.1.4.9: If the entries of A are ai j, use subscript notation to write (a) t...
 1.1.7.10: Compare the pivots in direct elimination to those with partial pivo...
 1.1.5.10: (a) Why does it take approximately n 2/2 multiplicationsubtraction...
 1.1.6.10: Find the inverses (in any legal way) of A1 = 0 0 0 1 0 0 2 0 0 3 0 ...
 1.1.2.10: Recommended) Under what condition on y1, y2, y3 do the points (0,y1...
 1.1.3.10: Reduce this system to upper triangular form by two row operations: ...
 1.1.4.10: True or false? Give a specific counterexample when false. (a) If co...
 1.1.7.11: Explain why partial pivoting produces multipliers `i j in L that sa...
 1.1.11: If E is 2 by 2 and it adds the first equation to the second, what a...
 1.1.5.11: Solve as two triangular systems, without multiplying LU to find A: ...
 1.1.6.11: Give examples of A and B such that (a) A+B is not invertible althou...
 1.1.2.11: These equations are certain to have the solution x = y = 0. For whi...
 1.1.3.11: Apply elimination (circle the pivots) and backsubstitution to solv...
 1.1.4.11: The first row of AB is a linear combination of all the rows of B. W...
 1.1.12: True or false, with reason if true or counterexample if false: (1) ...
 1.1.5.12: How could you factor A into a product UL, upper triangular times lo...
 1.1.6.12: If A is invertible, which properties of A remain true for A 1 ? (a)...
 1.1.2.12: Starting with x+4y = 7, find the equation for the parallel line thr...
 1.1.3.12: Which number d forces a row exchange, and what is the triangular sy...
 1.1.4.12: The product of two lower triangular matrices is again lower triangu...
 1.1.4.13: By trial and error find examples of 2 by 2 matrices such that (a) A...
 1.1.13: Solve Ax = b by solving the triangular systems Lc = b and Ux = c: A...
 1.1.5.13: Solve by elimination, exchanging rows when necessary: u + 4v + 2w =...
 1.1.6.13: If A is invertible, which properties of A remain true for A 1 ? (a)...
 1.1.2.13: Draw the two pictures in two planes for the equations x2y = 0, x+y = 6
 1.1.3.13: Which number b leads later to a row exchange? Which b leads to a mi...
 1.1.4.14: Describe the rows of EA and the columns of AE if E = " 1 7 0 1#
 1.1.14: If possible, find 3 by 3 matrices B such that (1) BA = 2A for every...
 1.1.5.14: Write down all six of the 3 by 3 permutation matrices, including P ...
 1.1.6.14: If B is square, show that A = B+B T is always symmetric and K = BB ...
 1.1.2.14: For two linear equations in three unknowns x, y, z, the row picture...
 1.1.3.14: (a) Construct a 3 by 3 system that needs two row exchanges to reach...
 1.1.4.15: Suppose A commutes with every 2 by 2 matrix (AB = BA), and in parti...
 1.1.15: Find the value for c in the following n by n inverse: if A = n 1 1 ...
 1.1.5.15: Find the PA = LDU factorizations (and check them) for A = 0 1 1 1 0...
 1.1.6.15: (a) How many entries can be chosen independently in a symmetric mat...
 1.1.2.15: For four linear equations in two unknowns x and y, the row picture ...
 1.1.3.15: If rows 1 and 2 are the same, how far can you get with elimination ...
 1.1.4.16: Let x be the column vector (1,0,...,0). Show that the rule (AB)x = ...
 1.1.16: For which values of k does kx + y = 1 x + ky = 1 have no solution, ...
 1.1.5.16: Find a 4 by 4 permutation matrix that requires three row exchanges ...
 1.1.6.16: (a) If A = LDU, with 1s on the diagonals of L and U, what is the co...
 1.1.2.16: Find a point with z = 2 on the intersection line of the planes x + ...
 1.1.3.16: Construct a 3 by 3 example that has 9 different coefficients on the...
 1.1.4.17: Which of the following matrices are guaranteed to equal (A+B) 2 ? A...
 1.1.17: Find the symmetric factorization A = LDLT of A = 1 2 0 2 6 4 0 4 11...
 1.1.5.17: The less familiar form A = LPU exchanges rows only at the end: A = ...
 1.1.6.17: If A = L1D1U1 and A = L2D2U2, prove that L1 = L2, D1 = D2, and U1 =...
 1.1.2.17: The first of these equations plus the second equals the third: x + ...
 1.1.3.17: Which number q makes this system singular and which righthand side...
 1.1.4.18: If A and B are n by n matrices with all entries equal to 1, find (A...
 1.1.18: Suppose A is the 4 by 4 identity matrix except for a vector v in co...
 1.1.5.18: Decide whether the following systems are singular or nonsingular, a...
 1.1.6.18: Under what conditions on their entries are A and B invertible? A = ...
 1.1.2.18: Move the third plane in to a parallel plane 2x + 3y + 2z = 9. Now t...
 1.1.3.18: (Recommended) It is impossible for a system of linear equations to ...
 1.1.4.19: A fourth way to multiply matrices is columns of A times rows of B: ...
 1.1.19: Solve by elimination, or show that there is no solution: u + v + w ...
 1.1.5.19: Which numbers a, b, c lead to row exchanges? Which make the matrix ...
 1.1.6.19: Compute the symmetric LDLT factorization of A = 1 3 5 3 12 18 5 18 ...
 1.1.2.19: In the columns are (1,1,2) and (1,2,3) and (1,1,2). This is a singu...
 1.1.3.19: Three planes can fail to have an intersection point, when no two pl...
 1.1.4.20: The matrix that rotates the xy plane by an angle is A() = " cos si...
 1.1.5.20: Forward elimination changes 1 1 1 2 x = b to a triangular 1 1 0 1 x...
 1.1.6.20: 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
 1.1.2.20: Normally 4 planes in fourdimensional space meet at a . Normally 4 ...
 1.1.3.20: Find the pivots and the solution for these four equations: 2x + y =...
 1.1.4.21: Find the powers A 2 , A 3 (A 2 times A), and B 2 , B 3 , C 2 , C 3 ...
 1.1.21: Describe the rows of DA and the columns of AD if D = [ 2 0 0 5 ].
 1.1.5.21: (Move to 3 by 3) Forward elimination changes Ax = b to a triangular...
 1.1.6.21: (Remarkable) If A and B are square matrices, show that I BA is inve...
 1.1.2.21: When equation 1 is added to equation 2, which of these are changed:...
 1.1.3.21: If you extend following the 1, 2, 1 pattern or the 1, 2, 1 pattern,...
 1.1.4.22: Write down the 3 by 3 matrices that produce these elimination steps...
 1.1.22: (a) If A is invertible what is the inverse of A T ? (b) If A is als...
 1.1.5.22: What are the 3 by 3 triangular systems Lc = b and Ux = c from 21? C...
 1.1.6.22: Find the inverses (directly or from the 2 by 2 formula) of A, B, C:...
 1.1.2.22: If (a,b) is a multiple of (c,d) with abcd 6= 0, show that (a,c) is ...
 1.1.3.22: Apply elimination and backsubstitution to solve 2u + 3v = 0 4u + 5...
 1.1.4.23: . In 22, applying E21 and then E32 to the column b = (1,0,0) gives ...
 1.1.23: By experiment with n = 2 and n = 3, find " 2 3 0 0#n , " 2 3 0 1#n ...
 1.1.5.23: What two elimination matrices E21 and E32 put A into upper triangul...
 1.1.6.23: Solve for the columns of A 1 = " x t y z# : " 10 20 20 50#"x y # = ...
 1.1.2.23: . In these equations, the third column (multiplying w) is the same ...
 1.1.3.23: For the system u + v + w = 2 u + 3v + 3w = 0 u + 3v + 5w = 2, what ...
 1.1.4.24: Which three matrices E21, E31, E32 put A into triangular form U? A ...
 1.1.24: Starting with a first plane u+2vw = 6, find the equation for (a) th...
 1.1.5.24: What three elimination matrices E21, E31, E32 put A into upper tria...
 1.1.6.24: Show that [ 1 2 3 6 ] has no inverse by trying to solve for the col...
 1.1.3.24: Solve the system and find the pivots when 2u v = 0 u + 2v w = 0 v +...
 1.1.4.25: Suppose a33 = 7 and the third pivot is 5. If you change a33 to 11, ...
 1.1.25: What multiple of row 2 is subtracted from row 3 in forward eliminat...
 1.1.5.25: When zero appears in a pivot position, A = LU is not possible! (We ...
 1.1.6.25: (Important) If A has row 1 + row 2 = row 3, show that A is not inve...
 1.1.3.25: Apply elimination to the system u + v + w = 2 3u + 3v w = 6 u v + w...
 1.1.4.26: If every column of A is a multiple of (1,1,1), then Ax is always a ...
 1.1.26: (a) What vector x will make Ax = column 1 of A + 2(column 3), for a...
 1.1.5.26: Which number c leads to zero in the second pivot position? A row ex...
 1.1.6.26: If A has column 1 + column 2 = column 3, show that A is not inverti...
 1.1.3.26: Solve by elimination the system of two equations x y = 0 3x + 6y = ...
 1.1.4.27: What matrix E31 subtracts 7 times row 1 from row 3? To reverse that...
 1.1.27: True or false, with reason if true and counterexample if false: (1)...
 1.1.5.27: What are L and D for this matrix A? What is U in A = LU and what is...
 1.1.6.27: Suppose A is invertible and you exchange its first two rows to reac...
 1.1.3.27: Find three values of a for which elimination breaks down, temporari...
 1.1.4.28: (a) E21 subtracts row 1 from row 2 and then P23 exchanges rows 2 an...
 1.1.28: By experiment or the GaussJordan method compute 1 0 0 ` 1 0 m 0 1 ...
 1.1.5.28: A and B are symmetric across the diagonal (because 4 = 4). Find the...
 1.1.6.28: If the product M = ABC of three square matrices is invertible, then...
 1.1.3.28: True or false: (a) If the third equation starts with a zero coeffic...
 1.1.4.29: (a) What 3 by 3 matrix E13 will add row 3 to row 1? (b) What matrix...
 1.1.29: Write down the 2 by 2 matrices that (a) reverse the direction of ev...
 1.1.5.29: (Recommended) Compute L and U for the symmetric matrix A = a a a a ...
 1.1.6.29: Prove that a matrix with a column of zeros cannot have an inverse.
 1.1.3.29: (Very optional) Normally the multiplication of two complex numbers ...
 1.1.4.30: Multiply these matrices: 0 0 1 0 1 0 1 0 0 1 2 3 4 5 6 7 8 9 0 0 1 ...
 1.1.5.30: Find L and U for the nonsymmetric matrix A = a r r r a b s s a b c ...
 1.1.6.30: Multiply [ a b c d ] times [ d b c a ]. What is the inverse of each...
 1.1.3.30: Use elimination to solve u + v + w = 6 u + 2v + 2w = 11 2u + 3v 4w ...
 1.1.4.31: This 4 by 4 matrix needs which elimination matrices E21 and E32 and...
 1.1.5.31: Tridiagonal matrices have zero entries except on the main diagonal ...
 1.1.6.31: (a) What matrix E has the same effect as these three steps? Subtrac...
 1.1.3.31: For which three numbers a will elimination fail to give three pivot...
 1.1.4.32: Write these ancient problems in a 2 by 2 matrix form Ax = b and sol...
 1.1.5.32: Solve the triangular system Lc = b to find c. Then solve Ux = c to ...
 1.1.6.32: Find the numbers a and b that give the inverse of 5 eye(4) ones(4,4...
 1.1.3.32: Find experimentally the average size (absolute value) of the first ...
 1.1.4.33: The parabola y = a+bx+cx2 goes through the points (x,y) = (1,4) and...
 1.1.5.33: Solve Lc = b to find c. Then solve Ux = c to find x. What was A? L ...
 1.1.6.33: Show that A = 4 eye(4) ones(4,4) is not invertible: Multiply A ones...
 1.1.4.34: Multiply these matrices in the orders EF and FE and E 2 : E = 1 0 0...
 1.1.5.34: If A and B have nonzeros in the positions marked by x, which zeros ...
 1.1.6.34: There are sixteen 2 by 2 matrices whose entries are 1s and 0s. How ...
 1.1.4.35: (a) Suppose all columns of B are the same. Then all columns of EB a...
 1.1.5.35: (Important) If A has pivots 2, 7, 6 with no row exchanges, what are...
 1.1.6.35: Change I into A 1 as you reduce A to I (by row operations): h A Ii ...
 1.1.4.36: If E adds row 1 to row 2 and F adds row 2 to row 1, does EF equal FE?
 1.1.5.36: Starting from a 3 by 3 matrix A with pivots 2, 7, 6, add a fourth r...
 1.1.6.36: Follow the 3 by 3 text example but with plus signs in A. Eliminate ...
 1.1.4.37: The first component of Ax is a1 jx j = a11x1 + + a1nxn. Write formu...
 1.1.5.37: Use chol(pascal(5)) to find the triangular factors of MATLABs pasca...
 1.1.6.37: Use GaussJordan elimination on [A I] to solve AA1 = I: 1 a b 0 1 c...
 1.1.4.38: If AB = I and BC = I, use the associative law to prove A = C.
 1.1.5.38: (Review) For which numbers c is A = LU impossiblewith three pivots?...
 1.1.6.38: Invert these matrices A by the GaussJordan method starting with [A...
 1.1.4.39: A is 3 by 5, B is 5 by 3, C is 5 by 1, and D is 3 by 1. All entries...
 1.1.5.39: Estimate the time difference for each new righthand side b when n ...
 1.1.6.39: Exchange rows and continue with GaussJordan to find A 1 : h A Ii =...
 1.1.4.40: What rows or columns or matrices do you multiply to find (a) the th...
 1.1.5.40: There are 12 even permutations of (1,2,3,4), with an even number of...
 1.1.6.40: True or false (with a counterexample if false and a reason if true)...
 1.1.4.41: (3 by 3 matrices) Choose the only B so that for every matrix A, (a)...
 1.1.5.41: How many exchanges will permute (5,4,3,2,1) back to (1,2,3,4,5)? Ho...
 1.1.6.41: For which three numbers c is this matrix not invertible, and why no...
 1.1.4.42: True or false? (a) If A 2 is defined then A is necessarily square. ...
 1.1.5.42: If P1 and P2 are permutation matrices, so is P1P2. This still has t...
 1.1.6.42: Prove that A is invertible if a 6= 0 and a 6= b (find the pivots an...
 1.1.4.43: If A is m by n, how many separate multiplications are involved when...
 1.1.5.43: (Try this question.) Which permutation makes PA upper triangular? W...
 1.1.6.43: This matrix has a remarkable inverse. Find A 1 by elimination on [A...
 1.1.4.44: To prove that (AB)C = A(BC), use the column vectors b1,...,bn of B....
 1.1.5.44: Find a 3 by 3 permutation matrix with P 3 = I (but not P = I). Find...
 1.1.6.44: If B has the columns of A in reverse order, solve (AB)x = 0 to show...
 1.1.4.45: Multiply AB using columns times rows: AB = 1 0 2 4 2 1 " 3 3 0 1 2 ...
 1.1.5.45: If you take powers of a permutation, why is some P k eventually equ...
 1.1.6.45: Find and check the inverses (assuming they exist) of these block ma...
 1.1.4.46: Block multiplication separates matrices into blocks (submatrices). ...
 1.1.5.46: The matrix P that multiplies (x,y,z) to give (z,x,y) is also a rota...
 1.1.6.46: Use inv(S) to invert MATLABs 4 by 4 symmetric matrix S = pascal(4)....
 1.1.4.47: Draw the cuts in A and B and AB to show how each of the four multip...
 1.1.5.47: If P is any permutation matrix, find a nonzero vector x so that (I ...
 1.1.6.47: If A = ones(4,4) and b = rand(4,1), how does MATLAB tell you that A...
 1.1.4.48: Block multiplication says that elimination on column 1 produces EA ...
 1.1.5.48: If P has 1s on the antidiagonal from (1,n) to (n,1), describe PAP.
 1.1.6.48: M1 shows the change in A 1 (useful to know) when a matrix is subtra...
 1.1.4.49: Elimination for a 2 by 2 block matrix: When A 1A = I, multiply the ...
 1.1.6.49: Find A T and A 1 and (A 1 ) T and (A T ) 1 for A = " 1 0 9 3# and a...
 1.1.4.50: With i 2 = 1, the product (A + iB)(x + iy) is Ax + iBx + iAy By. Us...
 1.1.6.50: Verify that (AB) T equals B TA T but those are different from A TB ...
 1.1.4.51: Suppose you solve Ax = b for three special righthand sides b: Ax1 ...
 1.1.6.51: (a) The matrix (AB) 1 T comes from (A 1 ) T and (B 1 ) T . In what ...
 1.1.4.52: If the three solutions in Question 51 are x1 = (1,1,1) and x2 = (0,...
 1.1.6.52: Show that A 2 = 0 is possible but A TA = 0 is not possible (unless ...
 1.1.4.53: Find all matrices A = " a b c d# that satisfy A " 1 1 1 1# = " 1 1 ...
 1.1.6.53: (a) The row vector x T times A times the column y produces what num...
 1.1.4.54: If you multiply a northwest matrix A and a southeast matrix B, what...
 1.1.6.54: When you transpose a block matrix M = [ A B C D ] the result is MT ...
 1.1.4.55: Write 2x+3y+z+5t = 8 as a matrix A (how many rows?) multiplying the...
 1.1.6.55: Explain why the inner product of x and y equals the inner product o...
 1.1.4.56: What 2 by 2 matrix P1 projects the vector (x,y) onto the x axis to ...
 1.1.6.56: If A = A T and B = B T , which of these matrices are certainly symm...
 1.1.4.57: Write the inner product of (1,4,5) and (x,y,z) as a matrix multipli...
 1.1.6.57: If A = A T needs a row exchange, then it also needs a column exchan...
 1.1.4.58: In MATLAB notation, write the commands that define the matrix A and...
 1.1.6.58: (a) How many entries of A can be chosen independently, if A = A T i...
 1.1.4.59: The MATLAB commands A = eye(3) and v = [3:5] produce the 3 by 3 ide...
 1.1.6.59: Suppose R is rectangular (m by n) and A is symmetric (m by m). (a) ...
 1.1.4.60: If you multiply the 4 by 4 allones matrix A = ones(4,4) and the co...
 1.1.6.60: Factor these symmetric matrices into A = LDLT . The matrix D is dia...
 1.1.4.61: Invent a 3 by 3 magic matrix M with entries 1,2,...,9. All rows and...
 1.1.6.61: Wires go between Boston, Chicago, and Seattle. Those cities are at ...
 1.1.6.62: Producing x1 trucks and x2 planes requires x1 + 50x2 tons of steel,...
 1.1.6.63: Ax gives the amounts of steel, rubber, and labor to produce x in 62...
 1.1.6.64: Here is a new factorization of A into triangular times symmetric: S...
 1.1.6.65: A group of matrices includes AB and A 1 if it includes A and B. Pro...
 1.1.6.66: If every row of a 4 by 4 matrix contains the numbers 0, 1, 2, 3 in ...
 1.1.6.67: Prove that no reordering of rows and reordering of columns can tran...
 1.1.6.68: A square northwest matrix B is zero in the southeast corner, below ...
 1.1.6.69: Compare tic; inv(A); toc for A = rand(500) and A = rand(1000). The ...
 1.1.6.70: I = eye(1000); A = rand(1000); B = triu(A); produces a random trian...
 1.1.6.71: Show that L 1 has entries j/i for i j (the 1, 2, 1 matrix has this ...
Solutions for Chapter 1: Matrices and Gaussian Elimination
Full solutions for Linear Algebra and Its Applications,  4th Edition
ISBN: 9780030105678
Solutions for Chapter 1: Matrices and Gaussian Elimination
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 1: Matrices and Gaussian Elimination includes 273 full stepbystep solutions. Since 273 problems in chapter 1: Matrices and Gaussian Elimination have been answered, more than 9856 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications, was written by and is associated to the ISBN: 9780030105678. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications,, edition: 4.

Absolute value of a complex number
The absolute value of the complex number z = a + b is given by ?a2+b2; also, the length of the segment from the origin to z in the complex plane.

Base
See Exponential function, Logarithmic function, nth power of a.

Component form of a vector
If a vector’s representative in standard position has a terminal point (a,b) (or (a, b, c)) , then (a,b) (or (a, b, c)) is the component form of the vector, and a and b are the horizontal and vertical components of the vector (or a, b, and c are the x, y, and zcomponents of the vector, respectively)

Deductive reasoning
The process of utilizing general information to prove a specific hypothesis

Dependent event
An event whose probability depends on another event already occurring

Distance (on a number line)
The distance between real numbers a and b, or a  b

Equation
A statement of equality between two expressions.

Finite sequence
A function whose domain is the first n positive integers for some fixed integer n.

Jump discontinuity at x a
limx:a  ƒ1x2 and limx:a + ƒ1x2 exist but are not equal

Leaf
The final digit of a number in a stemplot.

Linear function
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers

Multiplicative inverse of a matrix
See Inverse of a matrix

Phase shift
See Sinusoid.

Placebo
In an experimental study, an inactive treatment that is equivalent to the active treatment in every respect except for the factor about which an inference is to be made. Subjects in a blind experiment do not know if they have been given the active treatment or the placebo.

Quotient identities
tan ?= sin ?cos ?and cot ?= cos ? sin ?

Range (in statistics)
The difference between the greatest and least values in a data set.

RRAM
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the righthand end point of each subinterval.

Secant line of ƒ
A line joining two points of the graph of ƒ.

Sequence of partial sums
The sequence {Sn} , where Sn is the nth partial sum of the series, that is, the sum of the first n terms of the series.

Yscl
The scale of the tick marks on the yaxis in a viewing window.