Find a basis for the subspace of Mnxn consisting of all diagonal matrices.

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)

If you keep the same basis vectors but put them in a different order, the change of basis matrix B is a __ matrix. If you keep the basis vectors in order but change their lengths, B is a __ matrix.