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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...
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 parameter k such 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. λ1 = 2, with 2-eigenvector [0 1 1]T, λ2= 0, with 0-eigenvector [1 0 1]T, λ3 = -2, with -2 -eigenvector [1 1 0]T.
a) Find a diagonal matrix D and an invertible matrix P such that A = PDP-1. b) With P as in from part (a) find P-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 of B. (c) Find an eigenvector of B corresponding to its smallest eigenvalue.
5) Suppose A is an n × n matrix. Recall that null(A) is the dimension of the null space of A (i.e., the space of solutions to the equation AX = 0)
a) What is the exact relation between n, 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 that AX = 0 has a nontrivial solution if and only if AX = B does not have a solution for some n × 1 matrix B.
6) Matrix A has characteristic polynomial CA(x) = (x − 2)(x + 1)2.
a) The size of A is (circle the correct answer) (a) 3×3 (b) 2×2 (c) 4×4 (d) 2×3 (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.)
