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notes2-2

# notes2-2 - 1-2 1 4-7 3-2 6-5 It is a remarkable and...

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SECTION 2.2 MATRIX INVERSES If the matrix A is square and there is a matrix C such that CA = AC = I , then A is called invertible and C is called the inverse of A , which we write as A - 1 . A matrix that is not invertible is called singular. FACT. Products and transposes of invertible matrices are invertible. A bunch of algebra verifies the rule on page 119 for 2 × 2 matrices. EXAMPLE. Find the inverse of " 1 - 2 3 - 4 # .

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To find A - 1 in general, we need to solve AC = I . Unravel this to see that we need to perform row operations on the matrix [ A I ] to produce [ I A - 1 ]. EXAMPLE. If it exists, find the inverse of
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Unformatted text preview: 1-2 1 4-7 3-2 6-5 . It is a remarkable and remarkably useful fact that the three row operations can be performed using multiplication on the left by what are called elementary matrices . " 0 1 1 0 #" 1-2 3-4 # " 3 0 0 1 #" 1-2 3-4 # " 1 0 5 1 #" 1-2 3-4 # Notice that our method for ﬁnding A-1 in general only solves AC = I . But for A to be invertible, we also need CA = I . Use elementary matrices to investigate more carefully the existence of A-1 . HOMEWORK: SECTION 2.2...
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