M 346 - Fall 2011 Midterm #2 - Solutions

# M 346 - Fall 2011 Midterm #2 - Solutions - M346 Midterm...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: M346 Midterm Exam 2 Solution I. (10 points) Find a matrix with eigenvalues − 2+3 i and − 2 − 3 i , and corresponding eigenvectors ( i 1 ) and ( − i 1 ) . Ans: Let P = [ i − i 1 1 ] and D = [ − 2 + 3 i − 2 − 3 i ] , we have A = PDP- 1 = [ i − i 1 1 ][ − 2 + 3 i − 2 − 3 i ][ i − i 1 1 ]- 1 = 1 2 i [ − 2 i − 3 − 2 i − 3 − 2 + 3 i − 2 − 3 i ][ 1 i − 1 i ] = 1 2 i [ − 4 i − 6 i 6 i − 4 i ] = [ − 2 − 3 3 − 2 ] II. (10 points) Let A = [ 0 1 0 0 ] and B = [ 0 0 1 0 ] . Use de nition of exponentials of matrices (that is, e C = ∞ ∑ n =0 C n n ! for any square matric C ) to compute e A and e B . Show that e A e B ̸ = e B e A . Ans: Since A n = B n = O 2 for n ≥ 2 , we have e A = I 2 + A = [ 1 1 0 1 ] and e B = I 2 + B = [ 1 0 1 1 ] Therefore e A e B = [ 2 1 1 1 ] and e B e A = [ 1 1 1 2 ] . III. Suppose that x (0) = 0 and x (1) = 1 , and for n ≥ 2 , x ( n ) = − x ( n − 1) + 6 x ( n − 2) . 1. (5 points) Convert this to a 2 × 2 rst-order matrix problem....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

M 346 - Fall 2011 Midterm #2 - Solutions - M346 Midterm...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online