Q95E Consider a block matrix A=[A1100... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q95E (page 103)

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 ?

Short Answer

Expert verified

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

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

Step by step solution

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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91影视

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 ?

Short Answer

Step by step solution

Reducing A to identity theoretically

Step 2:Solving for A-1

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91影视

Consider a block matrix A=[A1100A22], where role="math" localid="1660371822794" A11andA12are square matrices. For which choices of A11andA22 is A invertible? In these cases, what is ?

Short Answer

Step by step solution

Reducing A to identity theoretically

Step 2:Solving for A-1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Pure Maths

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

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 ?