MATH 1502 Midterm: MATH 1502 GT Exam3 Solutions
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3 1 a + 1 (a) (15 points) find the determinant of a. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A is not invertible i det a = 0, i. e. , when a = 2. , a basis of r3, (a) (4 points) write the change-of-coordinates matrix p for the coordinate change from basis b to the standard basis; 1 0 1 (b) (8 points) nd the inverse p 1 of p ; 0 0 2 1 1 1 (c) (8 points) nd the coordinates [x]b and [y]b relative to basis b for x = . 0 0 9 (a) (6 points) compute the eigenvalues of a. = ( 9)(( 1)2 4) = ( 9)( 3)( + 1) = 0. The eigenvalues are 1 = 9, 2 = 3, and 3 = 1. (b) (8 points) find the corresponding eigenvectors. 0 (c) (6 points) write down matrices p (invertible) and d (diagonal) such that a = 5 (a) (4 points) show that x is orthogonal to y.
