Math 227 Sample Mid-term Exam
1. Given the matrix A =
1 1 find a matrix X such that X -1 AX = -1 3 (Hint: AX = XB, find X one column at a time.)
2 3 0 2
= B.
4 2 2 3 2. Let A = 1 1 -5 -4
0 2 -2 -2 0 2 -1 -1 0 -1 0 1 , det(A-xI) = (2-x)3 (1-x), (A-2I)2 =
Math 227 Solutions to Extra Problems on the Sample Final Exam 1. Suppose that A is an n n matrix, that v1 , v2 , v3 are eigenvectors corresponding to distinct eigenvalues 1 , 2 , 3 . Let v = v1 + v2 + v3 . Show that the minimal polynomial of A on v is (x
Math 227 Solutions to Sample Final Exam 3 -1 which has eigenvalues 2, 2, find a matrix X such that 1 1 X -1 AX is in Jordan form; use this to find a matrix B such that B 2 = A. We need a Jordan chain of length 2. Find a vector v Ker(A - 2I)2 , v Ker(A - 2
Math 227 Mid-term Solutions 1. Suppose that A is a 3 3 matrix, and that X is an invertible matrix such that X -1 AX = a b c d e f . Denote the columns of X by v1 , v2 , v3 . Show that U = Spancfw_v1 , v2 is A0 0 g invariant, that is, Au U for u U . 6 2 1