Let A = 221 252 122 , and let S be the matrix with column vectors s1 = x 0 x , s2 = y y

Compute each of the following: (a) 5A (b) 3B (c) iC (d) 2A B (e) A + 3CT (f) 3D 2E (g) D + E + F (h) the matrix G such that 2A+3B2G = 5(A+B) (i) the matrix H such that D + 2F + H = 4E (j) the matrix K such that KT + 3A 2B = 023

Compute each of the following: (a) D (b) 4BT (c) 2AT + C (d) 5E + D (e) 4AT 2BT + iC (f) 4E 3DT(g) (1 6i)F + i D (h) the matrix G such that 2A B + (1 i)CT = G + A B (i) the matrix H such that 3D3E +6I32H = 03 (j) the matrix K such that KT + FT = DT + ET

For Problems 34, let A =  1 1 2 3 14 , B = 2 1 3 512 4 6 2 , C = 1 1 2 , D = 2 2 3 , E =  2 i 1 + i i 2 + 4i , F =  i 1 3i 0 4 + i . In these problems, i denotes 1.se problems, i denotes 1. 3. For each item, decide whether or not the given expression is defined. For each item that is defined, compute the result. (a) AB (b) BC (c) C A (d) AT E (e) C D (f) CT AT (g) F2 (h) B DT (i) AT A (j) F E

For Problems 34, let A =  1 1 2 3 14 , B = 2 1 3 512 4 6 2 , C = 1 1 2 , D = 2 2 3 , E =  2 i 1 + i i 2 + 4i , F =  i 1 3i 0 4 + i . In these problems, i denotes 1.For each item, decide whether or not the given expression is defined. For each item that is defined, compute the result. (a) AC (b) DC (c) DB (d) AD (e) E F (f) AT B (g) C2 (h) E2 (i) ADT (j) ET A

Let A =  32 7 1 6 0 3 5 , B = 2 8 8 3 1 9 0 2 , and C =  6 1 1 5 . Compute ABC and CAB.

For Problems 69, determine Ac by computing an appropriate linear combination of the column vectors of A.A =  1 3 5 4 , c =  6 2

For Problems 69, determine Ac by computing an appropriate linear combination of the column vectors of A.

For Problems 69, determine Ac by computing an appropriate linear combination of the column vectors of A.A = 1 2 4 7 5 4 , c =  5 1

For Problems 69, determine Ac by computing an appropriate linear combination of the column vectors of A.A =  abcd e f gh , c = x y z w .

Suppose A is an m n matrix and C is an r s matrix. (a) What must the dimensions of a matrix B be in order for the product ABC to be defined? (b) Write an expression for the (i, j) element of ABC in terms of the elements of A, B, and C.

Find A2, A3, and A4 if (a) A =  1 1 2 3 . (b) A = 0 10 2 01 4 1 0 .

If A, B, and C are n n matrices, show that (a) (A + 2B)2 = A2 + 2AB + 2B A + 4B2. (b) (A + B + C)2 = A2 + B2 + C2 + AB + B A + AC + C A + BC + C B. (c) (A B)3 = A3 ABA B A2 + B2 A A2B + AB2 + BAB B3.

If A =  2 5 6 6 , calculate A2 and verify that A satisfies A2 + 4A + 18I2 = 02.

If A = 104 112 230 , calculate A2 and A3 and verify that A satisfies A3 + A 26I3 = 03.

Find numbers x, y, and z such that the matrix A = 1 x z 0 1 y 001 satisfies A2 + 0 1 0 0 0 1 000 = I3.

If A =  x 1 2 y , determine all values of x and y for which A2 = A.

The Pauli spin matrices 1, 2, and 3 are defined by 1 =  0 1 1 0 , 2 =  0 i i 0 , and 3 =  1 0 0 1 . Verify that they satisfy 12 = i3, 23 = i1, 31 = i2. If A and B are n n matrices, we define their commutator, denoted [A, B], by [A, B] = AB B A. Thus, [A, B] = 0 if and only if A and B commute. That is, AB = B A. Problems 1922 require the commutator.

If A =  1 1 2 1 , B =  3 1 4 2 , find [A, B].

If A1 =  1 0 0 1 , A2 =  0 1 0 0 , and A3 =  0 0 1 0 , compute all of the commutators [Ai, Aj], and determine which of the matrices commute.

If A1 = 1 2  0 i i 0 , A2 = 1 2  0 1 1 0 , and A3 = 1 2  i 0 0 i , verify that [A1, A2] = A3,[A2, A3] = A1,[A3, A1] = A2.

If A, B, and C are nn matrices, find [A,[B,C]] and prove the Jacobi identity [A,[B,C]] + [B,[C, A]] + [C,[A, B]] = 0.

Use the index form of the matrix product to prove properties (2.2.4) and (2.2.5)

Prove parts (1) and (2) of Theorem 2.2.23.

Prove property (2) of the identity matrix.

If A and B are n n matrices, prove that tr(AB) = tr(B A).

For Problems 2627, let A = 3 1 6 , B = 0 4 7 1 1 3 , C =  903 2 115 2 , D = 215 007 1 2 1

For Problems 2627, let A = 3 1 6 , B = 0 4 7 1 1 3 , C =  903 2 115 2 , D = 215 007 1 2 1

Let A = 221 252 122 , and let S be the matrix with column vectors s1 = x 0 x , s2 = y y y , and s3 = z 2z z , where x, y,z are constants. (a) Show that AS = [s1,s2, 7s3]. (b) Find all values of x, y,z such that ST AS = diag(1, 1, 7).

Let A = 1 4 0 4 70 0 05 , and let S be the matrix with column vectorss1 = 0 0 z ,s2 = 2x x 0 , and s3 = y 2y 0 , where x, y,z are constants. (a) Show that AS = [5s1, s2, 9s3]. (b) Find all values of x, y,z such that ST AS = diag(5, 1, 9).

A matrix that is a scalar multiple of In is called an n n scalar matrix. (a) Determine the 4 4 scalar matrix whose trace is 8. (b) Determine the 3 3 scalar matrix such that the product of the elements on the main diagonal is 343.

Prove that for each positive integer n, there is a unique scalar matrix whose trace is a given constant k. If A is an n n matrix, then the matrices B and C defined by B = 1 2 (A + AT ), C = 1 2 (A AT ) are referred to as the symmetric and skew-symmetric parts of A respectively. Problems 3236 investigate properties of B and C.

Use the properties of the transpose to show that B and C are symmetric and skew-symmetric, respectively.

Show that A = B + C. Together with the previous exercise, this shows that every n n matrix A can be written as the sum of a symmetric matrix and a skew-symmetric matrix.

Find B and C for the matrix A = 4 1 0 9 2 3 2 55

Find B and C for the matrix A = 1 5 3 3 24 7 2 6 .

(a) If A is an n n symmetric matrix, what are B and C?(b) If A is an n n skew-symmetric matrix, what are B and C?

Prove that if A is an n p matrix and D = diag(d1, d2,..., dn), then D A is the matrix obtained by multiplying the i-th row vector of A by di , where 1 i n.

Prove that if A is an m n matrix and D = diag(d1, d2,..., dn), then AD is the matrix obtained by multiplying the j-th column vector of A by dj , where 1 j n.

If A and B are n n symmetric matrices such that AB = B A, then AB is symmetric.

Use the properties of the transpose to prove that (a) AAT is a symmetric matrix. (b) (ABC)T = CT BT AT .

For Problems 4144, determine the derivative of the given matrix function.

For Problems 4144, determine the derivative of the given matrix function.

For Problems 4144, determine the derivative of the given matrix function.

For Problems 4144, determine the derivative of the given matrix function.A(t) =  et e2t t2 2et 4e2t 5t2  .

Let A = [ai j(t)] be an m n matrix function and let B = [bi j(t)] be an n p matrix function. Use the definition of matrix multiplication to prove that d dt (AB) = A d B dt + d A dt B.

For Problems 4649, determine  b a A(t)dt for the given matrix function.A(t) =  et et 2et 5et  , a = 0, b = 1.

For Problems 4649, determine  b a A(t)dt for the given matrix function.

For Problems 4649, determine  b a A(t)dt for the given matrix function.The matrix function A(t) in Problem 44, with a = 0 and b = 1.

For Problems 4649, determine  b a A(t)dt for the given matrix function.A(t) = e2t sin 2t t2 5 tet sec2 t 3t sin t , a = 0, b = 1.

Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function. For each element of the matrix  A(t)dt, an arbitrary constant of integration must be included, and the arbitrary constants for different elements should be different. In Problems 5054, evaluate the indefinite integral  A(t)dt for the given matrix function. You may assume that the constants of all indefinite integrations are zero.A(t) =  5 1 t2 + 1 e3t

Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function. For each element of the matrix  A(t)dt, an arbitrary constant of integration must be included, and the arbitrary constants for different elements should be different. In Problems 5054, evaluate the indefinite integral  A(t)dt for the given matrix function. You may assume that the constants of all indefinite integrations are zero.A(t) =  2t 3t2 .

Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function. For each element of the matrix  A(t)dt, an arbitrary constant of integration must be included, and the arbitrary constants for different elements should be different. In Problems 5054, evaluate the indefinite integral  A(t)dt for the given matrix function. You may assume that the constants of all indefinite integrations are zero.The matrix function A(t) in Problem 43.

Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function. For each element of the matrix  A(t)dt, an arbitrary constant of integration must be included, and the arbitrary constants for different elements should be different. In Problems 5054, evaluate the indefinite integral  A(t)dt for the given matrix function. You may assume that the constants of all indefinite integrations are zero.The matrix function A(t) in Problem 46.

Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function. For each element of the matrix  A(t)dt, an arbitrary constant of integration must be included, and the arbitrary constants for different elements should be different. In Problems 5054, evaluate the indefinite integral  A(t)dt for the given matrix function. You may assume that the constants of all indefinite integrations are zero.The matrix function A(t) in Problem 49.