Instructions: Show all of your work; your work must justify your answers.

[1] (6 points) Let A be an n x n matrix. If the equation A

[2] (8 points) Let B and C be 4 x 4 matrices such that det B = 2 and det C = -5, and let A be the matrix

[ 0 4 5 1 ] (a) Determine det (3(B[3] (6 points) The nullspace of an m x n matrix A (with real number entries) is a subset of^{-3}C^{2})^{T}). [ 1 0 7 1 ] [ 1 0 2 1 ] [ 0 1 3 0 ]. (b) Use Cramer's rule to find x_{2}in the equation Ax=b, whereb^{T}= ( 5, 7, 2, 3). (This can be done in your head, but show how you obtain your answer.) (c) Why can't you always use Cramer's rule to solve linear equations? Give an example of a linear system of equations for which you can't use Cramer's rule, and explain why Cramer's rule doesn't work for your example.

[4] (9 points) For this problem, you may assume that B is the reduced row echelon form of A, where A and B are the following matrices:

[ 3 1 -2 0 1 2 1 ] [ 1 0 -1 0 0 -2 -3 ] (a) Find a basis for the [ 1 1 0 -1 1 2 2 ] [ 0 1 1 0 0 2 3 ] column space of A. [ 3 2 -1 1 1 8 9 ] [ 0 0 0 1 0 4 5 ] [ 0 2 2 -1 1 6 8 ] [ 0 0 0 0 1 6 7 ] (b) Find a basis for the [ 0 3 3 3 -3 0 3 ] [ 0 0 0 0 0 0 0 ] null space of A.[5] (6 points) Find the coordinates of ( 1, 2)

[6] (6 points): Say A is a 5x7 matrix with dim Nul(A)=4. Find the dimension of the row space of A and of the null space of A

[7] (6 points): Let A be the matrix

[ 1 -4 3 ] [ 4 -4 0 ] [ 2 -4 2 ].In each case determine if the given vector is an eigenvector of A and if so find its eigenvalue.

- (a)
**v**= [1,1,1]^{T} - (b)
**v**= [1,0,1]^{T} - (c)
**v**= [0,0,0]^{T}

[ 1 1 1 ] [ 0 2 1 ] [ 0 0 3 ].This is a diagonalizable matrix; thus P

- (a) Find each eigenvalue l of A,
and a basis for each eigenspace E
_{l}. - (b) Find P and D.

(a) The 5x5 identity matrix. (b) [ 1 1 0 0 0 ] [ 0 1 1 0 0 ] [ 0 0 1 0 0 ] [ 0 0 0 0 1 ] [ 0 0 0 0 0 ] (c) A 5x5 matrix A with eigenvalues (d) A matrix of the form PDP[10] (8 points): Let A be the matrix^{-1}, where 0, 2, and 3, and such that the P is invertible and D is diagonal. dimension of the null space is 2 for both A and A-3I.

[ 0 -1 ] [ 2 3 ].

- (a) Find the eigenvalues of A.
- (b) Find a basis for the eigenspace corresponding to each eigenvalue of A.
- (c) Find A
^{10}**v**, where**v**= [1, 0]^{T}.