1. A matrix is symmetric if for each pair of indices i and j, the i, j entry equals the j, i entry. A matrix is antisymmetric if each i, j entry is the negative of the j, i entry. (a)Give a symmetric 2 x 2 matrix and an antisymmetric 2 x 2 matrix. (b)Show the set of all the symmetric 2 x 2 matrices is a linear space(denote SR²X2); (c) Give a basis of SR2x2 and the dim(SR²x2).

8) Let A be an nxn ma triX. vector b the equation Ax = b has more than one solution. Explain why A is not an invertible Matrix.

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)