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Show that the block upper triangular matrix $A$ in Example 5 is invertible if and only if both $A_{11}$ and $A_{22}$ are invertible. IHint: If $A_{11}$ and $A_{22}$ are invertible, the formula for $A^{-1}$ given in Example 5 actually works as the inverse of $A . ]$ This fact about $A$ is an important part of several computer algorithms that estimate eigenvalues of matrices. Eigenvalues are discussed in Chapter $5 .$

block upper triangular matrix $A$ is invertible if and only if both the block matrices $A _ { 11 }$ and $A _ { 22 }$ are invertible.

Algebra

Chapter 2

Matrix Algebra

Section 4

Partitioned Matrices

Introduction to Matrices

Missouri State University

Campbell University

McMaster University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

03:04

Use part (c) of the Invert…

01:58

Use the fact that if $\bol…

02:39

show that matrix $A$ is th…

05:36

Show that matrix $A$ is th…

01:18

If $A$ is a $2 \times 2$ m…

09:32

Let $A$ be an $n \times n$…

01:38

Use matrix algcbra to show…

05:38

The inverse of a $2 \times…

03:51

Use the equivalence of (a)…

04:03

Okay, so four problem. 14. I want to show that block block up a triangular matrix. Same example. Five convertible even. Only if Jose Won 1 82 aren't vertical. So here's the given matrix in, in example five. I have already set it down and by our example five. The result from our example. Five. It says, uh, the members of a ISS equal to such a matrix. It's a one y in purse and detective, a 11 in verse and a 12 a lot of times a 22 in verse zero a 22 members. So by observe destructor off the of the avian bird is keeping by our example we find out that only we only require we only require a 11 and a 22 our Ian Birbal and it is okay for a one to to be in an uber go because we don't we don't know what we don't need to calculate the checkmate, the inverse of a 12 As we use our eyes, we apply. It is formula. So that means if that means even if Metro's ese invert both, then they won't want a one a 22 are both in Birbal and vice versa. Because he's obviously this is a equivalent, uh, process as we as we did, or 44 as we as what we did in the example.

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