Solved: Let A be an n n matrix. Show that if A2 = O, then I A is nonsingular and (I A) 1

Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n n matrices A and B. (a) (a + b) 2 = a2 + 2ab + b2 (b) (a + b)(a b) = a2 b2

Will the rules in Exercise 1 work if a is replaced by an n n matrix A and b is replaced by the n n identity matrix I?

Find nonzero 2 2 matrices A and B such that AB = O.

Find nonzero matrices A, B, and C such that AC = BC and A = B

The matrix A = 1 1 1 1 has the property that A2 = O. Is it possible for a nonzero symmetric 2 2 matrix to have this property? Prove your answer

Prove the associative law of multiplication for 22 matrices; that is, let A = a11 a12 a21 a22 , B = b11 b12 b21 b22 , C = c11 c12 c21 c22 and show that (AB)C = A(BC)

Let A = 1 2 1 2 1 2 1 2 Compute A2 and A3 . What will An turn out to be?

Let A = 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Compute A2 and A3. What will A2n and A2n+1 turn out to be?

Let A = 0100 0010 0001 0000 Show that An = O for n 4.

Let A and B be symmetric n n matrices. For each of the following, determine whether the given matrix must be symmetric or could be nonsymmetric: (a) C = A + B (b) D = A2 (c) E = AB (d) F = ABA (e) G = AB + BA (f) H = AB BA

Let C be a nonsymmetric n n matrix. For each of the following, determine whether the given matrix must necessarily be symmetric or could possibly be nonsymmetric: (a) A = C + CT (b) B = C CT (c) D = CTC (d) E = CTC CCT (e) F = (I + C)(I + CT ) (f) G = (I + C)(I CT )

Let A = a11 a12 a21 a22 Show that if d = a11a22 a21a12 = 0, then A1 = 1 d a22 a12 a21 a11

Use the result from Exercise 12 to find the inverse of each of the following matrices: (a) 7 2 3 1 (b) 3 5 2 3 (c) 4 3 2 2

Let A and B are n n matrices. Show that if AB = A and B = I then A must be singular.

Let A be a nonsingular matrix. Show that A1 is also nonsingular and (A1) 1 = A.

Prove that if A is nonsingular then AT is nonsingular and (AT ) 1 = (A1 ) T Hint: (AB) T = BT AT .

Let A be an n n matrix and let x and y be vectors in Rn. Show that if Ax = Ay and x = y, then the matrix A must be singular.

Let A be a nonsingular n n matrix. Use mathematical induction to prove that Am is nonsingular and (Am) 1 = (A1 ) m for m = 1, 2, 3, ... .

Let A be an n n matrix. Show that if A2 = O, then I A is nonsingular and (I A) 1 = I + A.

Let A be an n n matrix. Show that if Ak+1 = O, then I A is nonsingular and (I A) 1 = I + A + A2 ++ Ak

Given R = cos sin sin cos show that R is nonsingular and R1 = RT

An n n matrix A is said to be an involution if A2 = I. Show that if G is any matrix of the form G = cos sin sin cos then G is an involution.

Let u be a unit vector in Rn (i.e., uT u = 1) and let H = I 2uuT . Show that H is an involution.

A matrix A is said to be idempotent if A2 = A. Show that each of the following matrices are idempotent. (a) 1 0 1 0 (b) 2 3 1 3 2 3 1 3 (c) 1 4 1 4 1 4 1 4 1 4 1 4 1 2 1 2 1 2

Let A be an idempotent matrix. (a) Show that I A is also idempotent. (b) Show that I + A is nonsingular and (I + A) 1 = I 1 2A

Let D be an n n diagonal matrix whose diagonal entries are either 0 or 1. (a) Show that D is idempotent. (b) Show that if X is a nonsingular matrix and A = XDX1, then A is idempotent.

Let A be an involution matrix and let B = 1 2 (I + A) and C = 1 2 (I A) Show that B and C are both idempotent and BC = O

Let A be an m n matrix. Show that AT A and AAT are both symmetric.

Let A and B be symmetric n n matrices. Prove that AB = BA if and only if AB is also symmetric

Let A be an n n matrix and let B = A + AT and C = A AT (a) Show that B is symmetric and C is skew symmetric. (b) Show that every n n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix

In Application 1, how many married women and how many single women will there be after 3 years?

Consider the matrix A = 01011 10110 01001 11001 10110 (a) Draw a graph that has A as its adjacency matrix. Be sure to label the vertices of the graph. (b) By inspecting the graph, determine the number of walks of length 2 from V2 to V3 and from V2 to V5. (c) Compute the second row of A3 and use it to determine the number of walks of length 3 from V2 to V3 and from V2 to V5.

Consider the graph V2 V3 V5 V1 V4 (a) Determine the adjacency matrix A of the graph. (b) Compute A2. What do the entries in the first row of A2 tell you about walks of length 2 that start from V1? (c) Compute A3. How many walks of length 3 are there from V2 to V4? How many walks of length less than or equal to 3 are there from V2 to V4? For each of the conditional statements that follow, answer true if the statement is always true and answer false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.

If Ax = Bx for some nonzero vector x, then the matrices A and B must be equal.

If A and B are singular n n matrices, then A + B is also singular.

If A and B are nonsingular matrices, then (AB) T is nonsingular and ((AB) T ) 1 = (A1 ) T (B1 )