(2.4) Prove that if X is an invertible matrix then X + Y and (In + YX−¹) are both invertible or not invertible. (2.5) Prove that if X, Y and X + Y are invertible matrices of the same size, then X(X¹+Y-¹) Y(X+Y)-¹ = 1. 1

Give an example of a 2 x 3 matrix A that is in RREF and whose solutions to 2 (3) 4 1 Ax = 0 is spanned by the vector a =

Let A B denotes the tensor product of matrices Amxn and Bpxq defined by the block matrix A B = a12B 922B a11B a21 B : : ami B am2 B : ain B a2n B : amn B mpxnq (a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr. Prove that (AB)(C > D) = ACⒸ BD (b) Let Amxm and B₁xn be nonsingular matrices. - Prove that (AB) is nonsingular. - Find the inverse matrix of (AB) (c) Prove that the following equality applies for any Amxm and Bnxn square matrices trace(AB) = trace(A) * trace(B)

Let A = [1] 0 a and B = -5 -4 b be two similar matrices. (a) Find a and b. (b) Find an invertible matrix C such that A = CBC-¹.