MATH 235 Final: MATH 235 UMass Amherst final-fall01-compressed
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Fall 2015: (20 points) (a) show that the characteristic polynomial of the matrix a = is ( 3)2( 9). (b) find a basis of r3 consisting of eigenvectors of a. 0 (c) find an invertible matrix p and a diagonal matrix d such that the matrix. P 1ap = d (d) let b be a 5 5 matrix with characteristic polynomial. ( 1)2( 2)( 3)( 4). Assume that the rank of b i is 3. Justify your answer: (20 points) the vectors v1 = (cid:18) 1. 3 (cid:19) are eigenvectors of the matrix a = (cid:18) . 6 . 4. 3 . 7 (cid:19). (a) the eigenvalue of v1 is. 8 (cid:19) in the basis {v1, v2}. (b) find the coordinates of (cid:18) 1 (c) compute a20(cid:18) 1 (d) as n gets larger, the vector an(cid:18) 1. Hint: the vector ax is in col(a) for every vector x. 1: (20 points) consider the following orthogonal basis of r3.
