Question:

1) Let A = [1 -2 1 1] [-1 2 0 1] [2 -4 1 0]

a) Find all solutions of the system A [x y z w]^T= [-2 6 -8]^T. b) Find rank(A) and the dimension of the null space of A, {X :AX=0}.

c) Find basis of row(A), the space spanned by the rows of A. d) Find basis of col(A), the space spanned by the columns of A. e) Find ker(A) ={X :AX=0}.

f)Show that the set of all transposes of the vectors of row(A) constitutes the orthogonal complement of ker(A).

2)If possible, find conditions on parameterksuch that the following system has no solutions, one solution, or infinitely many solutions. Solve the system when possible.

3x + 2y + z = 12

4x + y = 14

2x + 2y + 2z = k

3) Matrix A= [-1 -1 1] [-2 0 2] [-1 1 1] has the following eigenvalues and eigenvectors.₁= 2, with 2-eigenvector [0 1 1]^T,₂= 0, with 0-eigenvector [1 0 1]^T,₃= -2, with -2 -eigenvector [1 1 0]^T.

a)Find a diagonal matrixDand an invertible matrixPsuch thatA=PDP^-1. b)WithPas in from part (a) findP^-1. c) Find A¹⁰

4) Let B = [3 2 1] [0 1 0] [0 2 2]

(a) Find the characteristic polynomial of B. (b) List all eigenvalues ofB. (c) Find an eigenvector ofBcorresponding to its smallest eigenvalue.

5) SupposeAis annnmatrix. Recall that null(A) is the dimension of the null space ofA(i.e., the space of solutions to the equationAX= 0)

a)What is the exact relation betweenn,rank(A) and null(A) (circle the correct answer)?

(i) rank(A) + null(A) = n (ii) rank(A) + n = null(A) (iii) n rank(A) + null(A) (iv) n null(A) = rank(A) (v) None (vi) null(A) + n = rank(A) (vii) n null(A) = rank(A) (viii) n + null(A) + rank(A) (ix) Other.

b)Using your answer to (a), prove thatAX= 0 has a nontrivial solution if and only ifAX=Bdoes not have a solution for somen1 matrixB.

6)MatrixAhas characteristic polynomial C_A(x) = (x2)(x+ 1)².

a)The size of A is (circle the correct answer) (a) 33 (b) 22 (c) 44 (d) 23 (e) Don't know.

b)Can you conclude from the above information only that A is invertible, and why? (YES NO)

c)Can you conclude from the above information only that A is diagonalizable, and why? (YES NO)

d)Assuming A is diagonalizable, write a diagonal matrix that A is similar to. (No need to show your work.)