8 10 pts True or False a T F There exists a 2 x 2 matrix A s

8. (10 pts) True or False. (a) T F: There exists a 2 x 2 matrix A such that A^2 = 0 but A # 0, where 0 is the 2 x 2 zero matrix. (b) T F: There exists a 2 x 2 matrix A such that A^2 = I but A not equal to plus minus I, where I is the 2 x 2 identity matrix. (c) T F: If A and B are square matrices of the same size then A^2 + 2AB + B^2 = (A+ B)^2. (d) T F: Let A, B and C be matrices of the same size. If A is invertible and AB = AC, then B = C. (e) T F: Let A be an n x n matrix and I the identity matrix. If A^2 +2A + I = 0, then A^-1 = -2I - A. (f) T F: If A and B are invertible, then we have ((AB)^T)^-1 = (A^-1)^T (B^-1)^T (g) T F: Let A be a square matrix. Then tr(A) = tr(A^T). (h) T F: Let A and B be square matrices. Then tr(A + B) = tr(A) + tr(B). (i) T F: If A is of size 1 x 2, and B is of size 2 x 1. Then tr(AB) = tr(BA). (j) T F: Let A, B and C be matrices of size m x n, n x I and I x r, respectively. Then ABC is a matrix with size m x r.

Solution

c) is false

all the remaining are true