Let A =1 1 0       1 1 0       0 0 2 Find an orthogonal matrix P such that P^−1AP is diagonal.

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 =

If A and B are invertible matrices and c is any scalar, then which of the following does not hold? (A-1)-1 = A (A")-1 = (A¯1)" (cA)~1 = {A-1 (AB)¯1 = A-1B-1

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)