lecture 13, August 14, 2018.pdf

lecture 13, August 14, 2018.pdf - Finding matrix inverses...

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Finding matrix inverses, and more on the invertible matrix theorem Jacob Shapiro MATH1115 Lecture 13, August 14, 2018
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Matrix inversion algorithm I A I n E N · · · E 1 A E N · · · E 1 = R E N · · · E 1 . Example We use the algorithm discussed in the previous lecture to find the inverse of the matrix A = 1 2 3 2 5 3 1 0 8 . 1 2 3 1 0 0 2 5 3 0 1 0 1 0 8 0 0 1 1 2 3 1 0 0 0 1 - 3 - 2 1 0 0 - 2 5 - 1 0 1 1 2 3 1 0 0 0 1 - 3 - 2 1 0 0 0 - 1 - 5 2 1 1 2 3 1 0 0 0 1 - 3 - 2 1 0 0 0 1 5 - 2 - 1
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Matrix inversion algorithm Example (continued) 1 2 3 1 0 0 0 1 - 3 - 2 1 0 0 0 1 5 - 2 - 1 1 2 0 - 14 6 3 0 1 0 13 - 5 - 3 0 0 1 5 - 2 - 1 1 0 0 - 40 16 9 0 1 0 13 - 5 - 3 0 0 1 5 - 2 - 1 Therefore A - 1 = - 40 16 9 13 - 5 - 3 5 - 2 - 1 .
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Matrix inversion algorithm Example We determine whether the matrix A = 1 6 4 2 4 - 1 - 1 2 5 is invertible. 1 6 4 1 0 0 2 4 - 1 0 1 0 - 1 2 5 0 0 1 1 6 4 1 0 0 0 - 8 - 9 - 2 1 0 0 8 9 1 0 1 1 6 4 1 0 0 0 - 8 - 9 - 2 1 0 0 0 0 - 1 1 1 Continued row reductions will not affect the last row, so we conclude that the reduced row echelon form of A is not the identity. Hence A is not invertible.
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Theorem (invertible matrix theorem) Let A be an n × n matrix. Then the following statements are equivalent (TFAE). a. A is invertible. b. The homogeneous equation A x = 0 has only the trivial solution x = 0 . c. The reduced row echelon form of A is I n . d. A is expressible as a product of elementary matrices. We will add even more equivalent statements as we go.
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Theorem A system of linear equations has zero, one, or infinitely many solutions.
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  • Three '08
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