solutions9_2011

# solutions9_2011 - Math 152 Spring 2011 Solutions...

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Unformatted text preview: Math 152, Spring 2011 Solutions Assignment #9 1. Let T : R 2 → R 2 be the linear transformation defined by T ( x,y ) = (2 x- 3 y, 4 x- 5 y ). (a) Write the matrix A that corresponds to T . (b) Calculate the determinant of A . (c) Is A invertible? If yes, calculate its inverse. Solution: (a) Since T (1 , 0) = (2 , 4) and T (0 , 1) = (- 3 ,- 5), then A = C 2- 3 4- 5 D . (b) det A = 2 · (- 5)- 4 · (- 3) =- 10 + 12 = 2. (c) Since det A = 2 Ó = 0, A is invertible and the formula for inverse of 2 × 2 matrices gives: A- 1 = 1 2 C- 5 3- 4 2 D . 2. Consider the following 5 × 5 matrices: M = 1 3- 7 6 0 1 3- 3 0 0 2- 1 3 0 3 6 3 2 0 1 1 , N = 1 6 2 π 2- 1 √ 2 21 36 2 74 45 π 52 1 . (a) Compute det M . (b) Compute det N . (c) Compute det( N 2 M T N- 1 M 3 ). (d) If the matrices M and N had been previously entered into MAT- LAB, write the MATLAB command to compute the value found in part (c). 1 Solution: (a) Since M has a lot of zero entries, we can use several expansions along columns or rows. For instance, expanding first along the first column, then the last one, and finally the last row, we obtain...
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solutions9_2011 - Math 152 Spring 2011 Solutions...

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