1. (2 marks) Let A - 1 (a) Using inspection, find a linear combination of the columns of A equal to (b] Using inspection, find a linear combination of the columns of A equal to (c) Using inspection, find a linear combination of the columns of A equal to (d) Use the linear combinations found above to write down a matrix B such that AB - Is, where Is is the 3 x 4 identity matrix. 2. (4 marks] In each part below, evaluate AB and BA, when they are defined. -4 (2) A= 10 6 -325 B = -6 3 -1 10 2 (B is called a permatation matriz Can you ace why?) a. (3 marks] Let A, B 6 My,(2,) be given by Find AB, BA, and A" (recall that A] = AA). 4. (4 marks] Let Find all 2 x 2 matrices B =[ in May[R) wach that AB - B.A. 5. Let A be an axn matrix. Recall that /2 - AA. A matrix A is called idexpotent if A' = A. (a] (1 mark) Show that the matrix is idempotent. (b) (3 marks) Let A and B be n X n matrices with AB = A and BA = B. Prove that B is idempotent. Justify each step of your proof (you may use the various properties of matrix multiplication that were given in class). Remember that matrix multiplication is not commutative! (c) (4 marks) Suppose A 6 M.(F] is an idempotent matrix, and let I = I,. Prove that / - A is idempotent [recall that ] - A means ] + (-4), where -A is the additive inverse of A in Men(F]]. Justify each step of your proof. Since M.(F) is a vector space, vector space axioms and theorems may be used along with the properties of matrix multiplication (for example, -A = [-1)4). 6. (4 marks) The trace of the n * n matrix A, denoted Tr A, is the sum of the entries on the main diagonal: Let A and B be n x n matrices. Prove that Tr /18 - Tr BA (even though AB and B.A may not be equal)