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Quiz6Solutions

# Quiz6Solutions - Newberger Math 247 Spring 03 Quiz 2.1-2.3...

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Newberger Math 247 Spring 03 Quiz 2.1-2.3 name: 1. Find the inverse of A = 1 0 0 5 1 0 0 2 2 . 1 0 0 1 0 0 5 1 0 0 1 0 0 2 2 0 0 1 1 0 0 1 0 0 0 1 0 - 5 1 0 0 2 2 0 0 1 1 0 0 1 0 0 0 1 0 - 5 1 0 0 0 2 - 10 - 2 1 Thus A - 1 = 1 0 0 - 5 1 0 - 5 - 1 1 2 . 2. Suppose A is invertible and AB = 0 , where 0 denotes the zero matrix. Show that B = 0 . Since A is invertible, we have the matrix A - 1 to work with. Multiplying AB = 0 by A - 1 , we get A - 1 ( AB ) = A - 1 0 . Using the associative law, this becomes ( A - 1 A ) B = A - 1 0 . But A - 1 A = I and the right hand side is 0 , so we get B = 0 . 3. Suppose B = PAP - 1 , where A and P are invertible n × n matrices. Find B - 1 . The inverse of the product of invertible matrices is the product of the inverses in reverse order. B - 1 = ( PAP - 1 ) - 1 = ( P - 1 ) - 1
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