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Math 3013 Quiz 3 Solutions

# Math 3013 Quiz 3 Solutions - Math 3013 Linear Algebra Fall...

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Math 3013 Linear Algebra, Fall 2012 Quiz 3 Oct. 10, 2012 1. Determine if the following matrix is invertible or not. If it is invertible, ﬁnd its inverse. A = 1 - 1 - 2 2 - 3 - 5 - 1 3 5 Solution ± A I ² = 1 - 1 - 2 1 0 0 2 - 3 - 5 0 1 0 - 1 3 5 0 0 1 r 2+ r 3 -→ 1 - 1 - 2 1 0 0 1 0 0 0 1 1 - 1 3 5 0 0 1 r 1 r 2 1 0 0 0 1 1 1 - 1 - 2 1 0 0 - 1 3 5 0 0 1 r 2 - r 1 ,r 3+ r 1 1 0 0 0 1 1 0 - 1 - 2 1 - 1 - 1 0 3 5 0 1 2 ( - 1) × r 2 1 0 0 0 1 1 0 1 2 - 1 1 1 0 3 5 0 1 2 r 3 - 3 r 2 1 0 0 0 1 1 0 1 2 - 1 1 1 0 0 - 1 3 - 2 - 1 ( - 1) × r 3 1 0 0 0 1 1 0 1 2 - 1 1 1 0 0 1 - 3 2 1 r 2 - 2 r 3 1 0 0 0 1 1 0 1 0 5 - 3 - 1 0 0 1 - 3 2 1 2. Compute the LU factorization of A = 2 - 1 0 0 6 - 4 5 - 3 8 - 4 1 0 4 - 1 0 7 Solution A = LU can be written into 2 - 1 0 0 6 - 4 5 - 3 8 - 4 1 0 4 - 1 0 7 = 1 0 0 0 * 1 0 0 * * 1 0 * * * 1 * * * * 0 * * * 0 0 * * 0 0 0 * 1. The ﬁrst row of U . By using matrix multiplication and the ﬁrst row of A , we have 2 - 1 0 0 6 - 4 5 - 3 8 - 4 1 0 4 - 1 0 7 = 1 0 0 0 *

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Math 3013 Quiz 3 Solutions - Math 3013 Linear Algebra Fall...

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