Short Answer

Consider a block matrix $A = [\begin{matrix} A_{11} & 0 \\ 0 & A_{22}^{} \end{matrix}]$ , where role="math" localid="1660371822794" $A_{11} and A_{12}$ are square matrices. For which choices of $A_{11} and A_{22}$ is A invertible? In these cases, what is ?

The matrix A is invertible when $A_{11} and A_{22}$ are invertible.

The invertible matrix is, $A = [\begin{matrix} A_{11}^{- 1} & 0 \\ 0 & A_{22}^{- 1} \end{matrix}]$ .

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: Reducing A to identity theoretically

Consider $A = [\begin{matrix} A_{11} & 0 \\ 0 & A_{22} \end{matrix}]$ where role="math" localid="1660372094980" $A_{11} and A_{22}$ are square matrices. We want to find what choices of $A_{11} and A_{22}$ make A invertible and what is for these choices.

Obviously, A will be invertible only if $A_{11} and A_{22}$ are invertible.This is because if $A_{11} and A_{22}$ are 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, $A = [\begin{matrix} A_{11}^{- 1} & 0 \\ 0 & A_{22}^{- 1} \end{matrix}]$ .

Now, check it by finding the product of ${AA}^{- 1}$ .

$A A^{- 1} = [\begin{matrix} A_{11}^{} & 0 \\ 0 & A_{22}^{} \end{matrix}] [\begin{matrix} A_{11}^{- 1} & 0 \\ 0 & A_{22}^{- 1} \end{matrix}] = [\begin{matrix} A_{11} A_{11}^{- 1} & 0 \\ 0 & A_{22} A_{22}^{- 1} \end{matrix}] = [\begin{matrix} l_{n} & 0 \\ 0 & l_{n} \end{matrix}]$

Where the product is identity matrix.

Thus, $A^{- 1} = [\begin{matrix} A_{11}^{- 1} & 0 \\ 0 & A_{22}^{- 1} \end{matrix}]$ .

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Short Answer

Consider a block matrix $A = [\begin{matrix} A_{11} & 0 \\ 0 & A_{22}^{} \end{matrix}]$ , where role="math" localid="1660371822794" $A_{11} and A_{12}$ are square matrices. For which choices of $A_{11} and A_{22}$ is A invertible? In these cases, what is ?

Step by Step Solution

Step 1: Reducing A to identity theoretically

Step 2:Solving for A-1

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Linear Algebra With Applications

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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 ?

Step by Step Solution

Step 1: Reducing A to identity theoretically

Step 2:Solving for A-1

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Consider a block matrix $A = [\begin{matrix} A_{11} & 0 \\ 0 & A_{22}^{} \end{matrix}]$ , where role="math" localid="1660371822794" $A_{11} and A_{12}$ are square matrices. For which choices of $A_{11} and A_{22}$ is A invertible? In these cases, what is ?