Let A, B, C be n × n real matrices such that A = BC. Prove A is invertible if and only if B, C are.

Let U be the space of 3 x 3 symmetric matrices and V be the space of 3 x 3 skew symmetric matrices. Show that dim U + dim V = 3².

Prove that the matrix A inverse as a product of elementary matrices. = 2 1 1 0 0 1 0 0 3 0-1 0 1 is invertible and write its

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)