M 346 - Fall 2011 Midterm #2 - Solutions

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

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

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