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Find bases for the column space of A, row space of A, and null space of a. verify that rank/nullity theorem holds. ( A is 4x4) A= [[1,4,-1,1][3,11,-1,4][1,5,2,3][2,8,-2,2] Use cramers rule to find solution. If no solution explain why X1 +x2 -2x3 = -3 3*x1-2*x2+2*x3 = 9 6*x1 -7*x2-x3 = 4 Find adj(A) and use it to compute A^(-1) A= [[2,0,1][0,0,2][0,0,1]] Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for matrix A. A=[[-1,0,0,0][5,-2,0,0][0,3,1,0][2,0,1,1]] Suppose A is a square matrix with characteristic polynomial (x-3)^3(x-2)^2(x+1) What are dimensions of A What are eigenvalues of A Is A invertible What is largest possible dimension for eigenspace of A Prove that u cannot be an eigenvector associated with two distinct eigenvalues x1 and x2 of A. Let A be an N x N matrix that is nilpotent of order k = 2. That is, assume that there exists an integer k (where k = 2) such that Ak = 0 (the N x N matrix consisting of zeros). Prove that 0 is the only eigenvalue of A. Find the change of basis matrix from B2 to B1. B1 and B2 contain two column vectors (e.g. B1 has colum vector [2] and [1] [ 1] [1] B1= {[[2][1]], [[1][1]]} B2 = {[[5][7]], [[3][4]]} B1 and B2 contain 3 column vectors with 3 rows. B1= {[[-1][1][-1]], [[1][0][2]], [[-2][5][0]]} and B2= {[[-2][1][3]], [[2][0][1]], [[4][1][-1]]}. Find XB1 if XB2= [[-1][1][3]]B2. Fiknd an example that meets given specifications. A basis B of R3 such that [[3][1][-2]]B = [[1][2][5]].