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Linear Algebra With Applications
Found in: Page 103
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Consider a block matrix A=[A1100A22 ], where role="math" localid="1660371822794" A11 and A12are square matrices. For which choices of A11 and A22 is A invertible? In these cases, what is ?

The matrix A is invertible when A11 and A22are invertible.

The invertible matrix is, A=A11-100A22-1.

See the step by step solution

Step by Step Solution

Step 1: Reducing A to identity theoretically

Consider A=A1100A22where role="math" localid="1660372094980" A11 and A22are square matrices. We want to find what choices of A11 and A22make A invertible and what is for these choices.

Obviously, A will be invertible only if A11 and A22are invertible.This is because if A11 and A22are invertible they can reduce to the identity which means that A can reduce to the identity as well.

Step 2:Solving for A-1

Logically this means that, A=A11-100A22-1.

Now, check it by finding the product of AA-1.

AA-1=A11 00A22 A11-100A22-1 =A11A11-100A22A22-1 =ln00ln

Where the product is identity matrix.

Thus, A-1= A11-100A22-1.

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