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ila0205 - 81 2.5 Inverse Matrices 2.5 Inverse Matrices...

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2.5. Inverse Matrices 81 2.5 Inverse Matrices Suppose A is a square matrix. We look for an “ inverse matrix A 1 of the same size, such that A 1 times A equals I . Whatever A does, A 1 undoes. Their product is the identity matrix—which does nothing to a vector, so A 1 A x D x . But A 1 might not exist . What a matrix mostly does is to multiply a vector x . Multiplying A x D b by A 1 gives A 1 A x D A 1 b . This is x D A 1 b . The product A 1 A is like multiplying by a number and then dividing by that number. A number has an inverse if it is not zero— matrices are more complicated and more interesting. The matrix A 1 is called “ A inverse.” DEFINITION The matrix A is invertible if there exists a matrix A 1 such that A 1 A D I and AA 1 D I: (1) Not all matrices have inverses . This is the first question we ask about a square matrix: Is A invertible? We don’t mean that we immediately calculate A 1 . In most problems we never compute it! Here are six “notes” about A 1 . Note 1 The inverse exists if and only if elimination produces n pivots (row exchanges are allowed). Elimination solves A x D b without explicitly using the matrix A 1 . Note 2 The matrix A cannot have two different inverses. Suppose BA D I and also AC D I . Then B D C , according to this “proof by parentheses”: B.AC/ D .BA/C gives BI D IC or B D C: (2) This shows that a left-inverse B (multiplying from the left) and a right-inverse C (multi- plying A from the right to give AC D I ) must be the same matrix . Note 3 If A is invertible, the one and only solution to A x D b is x D A 1 b : Multiply A x D b by A 1 : Then x D A 1 A x D A 1 b : Note 4 (Important) Suppose there is a nonzero vector x such that A x D 0 . Then A cannot have an inverse. No matrix can bring 0 back to x . If A is invertible, then A x D 0 can only have the zero solution x D A 1 0 D 0 . Note 5 A 2 by 2 matrix is invertible if and only if ad bc is not zero: 2 by 2 Inverse: a b c d 1 D 1 ad bc d b c a : (3) This number ad bc is the determinant of A . A matrix is invertible if its determinant is not zero (Chapter 5). The test for n pivots is usually decided before the determinant appears.

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82 Chapter 2. Solving Linear Equations Note 6 A diagonal matrix has an inverse provided no diagonal entries are zero: If A D 2 6 4 d 1 : : : d n 3 7 5 then A 1 D 2 6 4 1=d 1 : : : 1=d n 3 7 5 : Example 1 The 2 by 2 matrix A D 1 2 1 2 is not invertible. It fails the test in Note 5, because ad bc equals 2 2 D 0 . It fails the test in Note 3, because A x D 0 when x D .2; 1/ . It fails to have two pivots as required by Note 1. Elimination turns the second row of this matrix A into a zero row. The Inverse of a Product AB For two nonzero numbers a and b , the sum a C b might or might not be invertible. The numbers a D 3 and b D 3 have inverses 1 3 and 1 3 . Their sum a C b D 0 has no inverse.
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ila0205 - 81 2.5 Inverse Matrices 2.5 Inverse Matrices...

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