solutions2bd

solutions2bd - -6-7 2-3 2 6 3 2-1-1 2-1 2 . 2. (a) is...

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Math 33a/2, Quiz 2bd, October 18, 2007 Name: UCLA ID: 1. Compute the inverse of the matrix 1 2 5 - 2 - 3 - 6 0 - 1 - 6 . 2. Which of the following statements are true? (You need not explain your reasoning here; just write down the answers.) (a) If A and B are invertible 2 by 2 matrices, then A + B must be invertible as well. (b) If A 2 is an invertible 2 by 2 matrix, then A must be invertible as well. (c) If A and B are 2 by 2 matrices which represent reflections across lines L A and L B respectively, then the product AB must also represent reflection across a line. (Assume that L A and L B are lines through the origin.) Solution. 1. We perform row operations on the augmented matrix 1 2 5 | 1 0 0 - 2 - 3 - 6 | 0 1 0 0 - 1 - 6 | 0 0 1 +2(I) 1 2 5 | 1 0 0 0 1 4 | 2 1 0 0 - 1 - 6 | 0 0 1 -2(II) +(II) 1 0 - 3 | - 3 - 2 0 0 1 4 | 2 1 0 0 0 - 2 | 2 1 1 · - 1 2 1 0 - 3 | - 3 - 2 0 0 1 4 | 2 1 0 0 0 1 | - 1 - 1 2 - 1 2 +3(III) -4(III) 1 0 0 | - 6 - 7 2 - 3 2 0 1 0 | 6 3 2 0 0 1 | - 1 - 1 2 - 1 2 Hence the inverse matrix is
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Unformatted text preview: -6-7 2-3 2 6 3 2-1-1 2-1 2 . 2. (a) is false. For example, if A = I and B =-I , then both A and B are invertible, but A + B = 0, which is not invertible. (b) is true. Suppose A 2 is invertible and has inverse matrix B . Then A 2 B = I , by definition. Hence A ( AB ) = I . Therefore, A is invertible and has inverse AB . (c) is false. For example, if L A and L B happen to be the same line, then AB is the result of reflecting twice across the same line, which is just the identity transformation. The identity transformation certainly is not a reflection across any line. (More generally, the 1 product of two reflections will be a rotation; see problem 42 in section 2.4 of the Bretscher text.) 2...
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This note was uploaded on 04/02/2008 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.

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solutions2bd - -6-7 2-3 2 6 3 2-1-1 2-1 2 . 2. (a) is...

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