Which of the following are subspaces of Mnxn(R)? • ) The set of all symmetric matrices A in Mpxn(R). (i1) The set of all anti-symmetric matrices A in Mxn(IR). • (i) The set of all invertible matrices A in Mnxn(R). • (iv) The set of all matrices in M,nxn(R) with trace 0. (v) The set of all idempotent matrices A in Mnxn(R), that is A? = 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)

Find a basis for the row space of the matrix 1 1 0 2 2 2 -1 -2 A - -1 2 2 0

Show that a lower-triangular matrix L is invertible if and only if its main diagonal entries are nonzero.