Let A be a nonsingular nn matrix and let B be an n r matrix. Show that the reduced row

Which of the matrices that follow are elementary matrices? Classify each elementary matrix by type. (a) 0 1 1 0 (b) 2 0 0 3 (c) 1 0 0 0 1 0 5 0 1 (d) 1 0 0 0 5 0 0 0 1

Find the inverse of each matrix in Exercise 1. For each elementary matrix, verify that its inverse is an elementary matrix of the same type.

For each of the following pairs of matrices, find an elementary matrix E such that EA = B: (a) A = 2 1 5 3 , B = 4 2 5 3 (b) A = 2 1 3 2 4 5 3 1 4 , B = 2 1 3 3 1 4 2 4 5 (c) A = 4 2 3 1 0 2 2 3 1 , B = 4 2 3 1 0 2 0 3 5

For each of the following pairs of matrices, find an elementary matrix E such that AE = B: (a) A = 4 1 3 2 1 4 1 3 2 , B = 3 1 4 4 1 2 2 3 1 (b) A = 2 4 1 6 , B = 2 2 1 3 (c) A = 4 2 3 2 4 2 6 1 2 , B = 2 2 3 1 4 2 3 1 2

Let A = 1 2 4 2 1 3 1 0 2 , B = 1 2 4 2 1 3 2 2 6 , C = 1 2 4 0 1 3 2 2 6 (a) Find an elementary matrix E such that EA = B. (b) Find an elementary matrix F such that FB = C. (c) Is C row equivalent to A? Explain.

Let A = 2 1 1 6 4 5 4 1 3 (a) Find elementary matrices E1, E2, E3 such that E3E2E1A = U where U is an upper triangular matrix. (b) Determine the inverses of E1, E2, E3 and set L = E1 1 E1 2 E1 3 . What type of matrix is L? Verify that A = LU.

Let A = 2 1 6 4 (a) Express A as a product of elementary matrices. (b) Express A1 as a product of elementary matrices.

Compute the LU factorization of each of the following matrices: (a) 3 1 9 5 (b) 2 4 2 1 (c) 1 1 1 3 5 6 2 2 7 (d) 2 1 2 4 1 2 6 3 4

Let A = 1 0 1 3 3 4 2 2 3 (a) Verify that A1 = 1 2 3 1 1 1 0 2 3 (b) Use A1 to solve Ax = b for the following choices of b: (i) b = (1, 1, 1)T (ii) b = (1, 2, 3)T (iii) b = (2, 1, 0)T 1

Find the inverse of each of the following matrices: (a) 1 1 1 0 (b) 2 5 1 3 (c) 2 6 3 8 (d) 3 0 9 3 (e) 1 1 1 0 1 1 0 0 1 (f) 2 0 5 0 3 0 1 0 3 (g) 1 3 3 2 6 1 3 8 3 (h) 1 0 1 1 1 1 1 2 3 1

Given A = 3 1 5 2 and B = 1 2 3 4 compute A1 and use it to (a) find a 2 2 matrix X such that AX = B. (b) find a 2 2 matrix Y such that YA = B. 1

Let A = 5 3 3 2 , B = 6 2 2 4 ,C = 4 2 6 3 Solve each of the following matrix equations: (a) AX + B = C (b) XA + B = C (c) AX + B = X (d) XA + C = X 1

Is the transpose of an elementary matrix an elementary matrix of the same type? Is the product of two elementary matrices an elementary matrix? 1

Let U and R be nn upper triangular matrices and set T = UR. Show that T is also upper triangular and that t j j = u j jr j j for j = 1, . . . , n. 1

Let A be a 3 3 matrix and suppose that 2a1 + a2 4a3 = 0 How many solutions will the system Ax = 0 have? Explain. Is A nonsingular? Explain. 1

Let A be a 3 3 matrix and suppose that a1 = 3a2 2a3 Will the system Ax = 0 have a nontrivial solution? Is A nonsingular? Explain your answers. 1

Let A and B be n n matrices and let C = A B. Show that if Ax0 = Bx0 and x0 _= 0, then C must be singular. 1

Let A and B be n n matrices and let C = AB. Prove that if B is singular, then C must be singular. [Hint: Use Theorem 1.5.2.] 1

Let U be an n n upper triangular matrix with nonzero diagonal entries. (a) Explain why U must be nonsingular. (b) Explain why U1 must be upper triangular. 2

Let A be a nonsingular nn matrix and let B be an n r matrix. Show that the reduced row echelon form of (A|B) is (I |C), where C = A1B. 2

In general, matrix multiplication is not commutative (i.e., AB _= BA). However, in certain special cases the commutative property does hold. Show that (a) if D1 and D2 are n n diagonal matrices, then D1D2 = D2D1. (b) if A is an n n matrix and B = a0 I + a1A + a2A2 + +ak Ak where a0, a1, . . . , ak are scalars, then AB = BA. 2

Show that if A is a symmetric nonsingular matrix, then A1 is also symmetric. 2

Prove that if A is row equivalent to B, then B is row equivalent to A. 2

(a) Prove that if A is row equivalent to B and B is row equivalent to C, then A is row equivalent to C. (b) Prove that any two nonsingular n n matrices are row equivalent. 2

Let A and B be m n matrices. Prove that if B is row equivalent to A and U is any row echelon form A, then B is row equivalent to U. 2

Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B = MA. 2

Is it possible for a singular matrix B to be row equivalent to a nonsingular matrix A? Explain. 2

Given a vector x Rn+1, the (n + 1) (n + 1) matrix V defined by vi j = _ 1 ifj = 1 x j1 i for j = 2, . . . , n + 1 is called the Vandermonde matrix. (a) Show that if Vc = y and p(x) = c1 + c2x + +cn+1xn then p(xi ) = yi , i = 1, 2, . . . , n + 1 (b) Suppose that x1, x2, . . . , xn+1 are all distinct. Show that if c is a solution to Vx = 0, then the coefficients c1, c2, . . . , cn must all be zero and hence V must be nonsingular. 2

If A is row equivalent to I and AB = AC, then B must equal C. 3

If E and F are elementary matrices and G = EF, then G is nonsingular. 3

If A is a 4 4 matrix and a1 +a2 = a3 +2a4, then A must be singular. 3

If A is row equivalent to both B and C, then A is row equivalent to B + C.