Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Find the inverses (directly or from the 2 by 2 formula) of A, B, C: A = [~ ~ ] and B = [~ ~ ] and C = [; ~]

Introduction to Linear Algebra 4
For these "permutation matrices" find p-l by trial and error (with 1 's and O's): [ 0 0 1] P = 0 1 0 100 [ 0 1 0] and P = 0 0 1 . 100

Introduction to Linear Algebra 4
Solve for the first column (x, y) and second column (t, z) of A-I: and [ 10 20] [t] = [0] 20 50 z l

Introduction to Linear Algebra 4
Show that U ~] is not invertible by trying to solve AA -1 = I for column 1 of A-I: ( For a different A, could column 1 of A-I) be possible to find but not column 2?

Introduction to Linear Algebra 4
Find an upper triangular V (not diagonal) with V 2 = I which gives V = V-I.

Introduction to Linear Algebra 4
(a) If A is invertible and AB = AC, prove quickly that B = C. (b) If A = [11], find two different matrices such that A B = A C .

Introduction to Linear Algebra 4
(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 sides (bI , b2 , b3) might allow a solution to Ax = b? (c) What happens to row 3 in elimination?

Introduction to Linear Algebra 4
If A has column 1 + column 2 = column 3, show that A is not invertible: " (a) Find a nonzero solution x to Ax = O. The matrix is 3 by 3. (b) Elimination keeps column 1 + column 2 = column 3. Explain why there is no third pivot.

Introduction to Linear Algebra 4
Suppose A is invertible and you exchange its first two rows to reach B. Is the new matrix B invertible and how would you find B-1 from A-I?

Introduction to Linear Algebra 4
Find the inverses (in any legal way) of 0 0 0 2 3 2 0 0 A= 0 0 3 0 and B= 4 3 0 0 0 4 0 0 0 0 6 5 5 0 0 0 0 0 7 6

Introduction to Linear Algebra 4
(a) Find invertible matrices A and B such that A + B is not invertible. (b) Find singular matrices A and B such that A + B is invertible.

Introduction to Linear Algebra 4
If the product C = A B is invertible (A and B are square), then A itself is invertible. Find a formula for A-I that involves C -1 and B.

Introduction to Linear Algebra 4
If the product M = ABC ofthree square matrices is invertible, then B is invertible. (So are A and C.) Find a formula for B- 1 that involves M-l and A and C

Introduction to Linear Algebra 4
If you add row 1 of A to row 2 to get B, how do you find B- 1 from A-I? Notice the order. The inverse of B = [~ ~] [ A] is

Introduction to Linear Algebra 4
Prove that a matrix with a column of zeros cannot have an inverse.

Introduction to Linear Algebra 4
Multiply [~ ~] times [_~ -~]. What is the inverse of each matrix if ad =f:. be?

Introduction to Linear Algebra 4
(a) What 3 by 3 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

Introduction to Linear Algebra 4
If B is the inverse of A 2 , show that A B is the inverse of A.

Introduction to Linear Algebra 4
Find the numbers a and b that give the inverse of 5 * eye(4) - ones(4,4): 4 -1 -1 -1 -1 a b b b -1 4 -1 -1 b a b b -1 -1 4 - -1 b b a b -1 -1 -1 4 b b b a What are a and b in the inverse of 6 * eye(5) - ones(5,5)?

Introduction to Linear Algebra 4
Show that A = 4 * eye(4) - ones(4,4) is not invertible: Multiply A * ones(4, 1).

Introduction to Linear Algebra 4
There are sixteen 2 by 2 matrices whose entries are l's and O's. How many of them are invertible?

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
Questions 22-28 are about the Gauss-Jordan method for calculating A-I

Introduction to Linear Algebra 4
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) Every matrix with 1 's down the main diagonal is invertible. (c) If A is invertible then A-I and A2 are invertible.

Introduction to Linear Algebra 4
For which three numbers C is this matrix not invertible, and why not? [ 2 C C]

Introduction to Linear Algebra 4
Prove that A is invertible if a =1= 0 and a =1= b (find the pivots or A-I):

Introduction to Linear Algebra 4
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; then multiply to confirm. I -I Invert A = o I o 0 o 0 I -I -I I I -I o I and solve Ax = (1,1, 1, 1).

Introduction to Linear Algebra 4
Suppose the matrices P and Q have the same rows as I but in any order. They are "permutation matrices". Show that P - Q is singular by solving (P - Q)x = O.

Introduction to Linear Algebra 4
Find and check the inverses (assuming they exist) of these block matrices: [~ ~] [~ ~] [~ ].

Introduction to Linear Algebra 4
Could a 4 by 4 matrix A be invertible if every row contains the numbers 0,1,2,3 in some order? What if every row of B contains 0,1,2, -3 in some order?

Introduction to Linear Algebra 4
In the Worked Example 2.5 C, the triangular Pascal matrix L has an inverse with "alternating diagonals". Check that this L -1 is DLD, where the diagonal matrix D has alternating entries 1, -1,1, -1. Then LDLD = I, so what is the inverse of LD = pascal (4,1)?

Introduction to Linear Algebra 4
The Hilbert matrices have Hij = Ij(i + j - 1). Ask MATLAB for the exact 6 by 6 inverse invhilb(6). Then ask it to compute inv(hilb(6)). How can these be different, when the computer never makes mistakes

Introduction to Linear Algebra 4
a) Use inv(P) to invert MATLAB's 4 by 4 symmetric matrix P = pascal(4). (b) Create Pascal's lower triangular L = abs(pascal(4,1)) and test P = LLT.

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

Introduction to Linear Algebra 4
(Recommended) A is a 4 by 4 matrix with 1 's on the diagonal and -a, -b, -c on the diagonal above. Find A -1 for this bidiagonal matrix.

Introduction to Linear Algebra 4
Suppose E1, E2, E3 are 4 by 4 identity matrices, except E1 has a, b, c in column 1 and E2 has d, e in column 2 and E3 has f in column 3 (below the 1 's). Multiply L = E1E2E3 to show that all these nonzeros are copied into L. E1E2E3 is in the opposite order from elimination (because E3 is acting first

Introduction to Linear Algebra 4
Direct multiplications 1-4 give M M-I - I, and I would recommend doing #3. M-I shows the change in A-I (useful to know) when a matrix is subtracted from A: 1 M = I -uvT 2 M = A-uvT 3 M = I -VV 4 M=A-VW-IV and and and and M-I = I + uvT /(1- vTu) (rank 1 change in I) M-I = A-I + A-1uvT A-I /(1- vT A-Iu

Introduction to Linear Algebra 4
Second difference matrices have beautiful inverses if they start with TIl - 1 (instead of Kil = 2). Here is the 3 by 3 tridiagonal matrix T and its inverse: Tn = 1 [ 3 2 T-I = 2 2 1 1 l] One approach is Gauss-Jordan elimination on [T I]. That seems too mechanical. I would rather write T as the produc

Introduction to Linear Algebra 4
Here are two more difference matrices, both important. But are they invertible? 2 -1 Cyclic C = -1 2 0 -1 -1 0 0 -1 2 -1 -1 0 -1 2 Free ends F = 1 -1 0 0 -1 2 -1 0 o -1 2-1 o 0 -1 1 One test is elimination-the fourth pivot fails. Another test is the detenninant, we don't want that. The best way is mu

Introduction to Linear Algebra 4
Elimination for a 2 by 2 block matrix: When you multiply the first block row by CA- 1 and subtract from the second row, the "Schur complement" S appears: A and D are square S = D -CA-1B. Multiply on the right to subtract A-I B times block column 1 from block column 2.

Introduction to Linear Algebra 4
How does the identity A(J + BA) = (J + AB)A connect the inverses of I + BA and I + AB? Those are both invertible or both singular: not obvious.