mat22a-midterm2-sol

mat22a-midterm2-sol - Math 22A UC Davis, Winter 2011 Prof....

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Math 22A UC Davis, Winter 2011 Prof. Dan Romik Midterm Exam 2 Solutions 1. For each of the following matrices, determine if it is invertible, and if it is, find its inverse matrix. (a) ± 1 2 3 4 ² (c) 1 1 1 0 2 - 2 1 3 - 1 (b) 1 0 2 0 4 - 2 0 0 0 (d) 1 1 1 0 1 2 0 1 3 Solution. (a) Recall that the inverse of a 2 × 2 matrix ± a b c d ² exists if its determinant ad - bc 6 = 0, and in that case the inverse matrix is equal to 1 ad - bc ± d - b - c a ² . In this case ad - bc = 4 - 6 = - 2 so the inverse matrix exists and is ± - 2 1 3 2 - 1 2 ² . (b) This matrix has a row of zeroes so its determinant is 0 and therefore it is not invertible. (c) The determinant of this matrix is also 0 so it is not invertible. (d) By applying elementary row operations to bring the matrix to reduced row echelon form and performing the same operations in parallel on the identity matrix, we get 1 1 1 1 0 0 0 1 2 0 1 0 0 1 3 0 0 1 1 0 - 1 1 - 1 0 0 1 2 0 1 0 0 0 1 0 - 1 1 1 0 0
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mat22a-midterm2-sol - Math 22A UC Davis, Winter 2011 Prof....

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