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Suppose the block matrix $A$ on the left side of $(7)$ is invertible and $A_{11}$ is invertible. Show that the Schur complement $S$ of $A_{11}$ is invertible. [Hint: The outside factors on the right side of $(7)$ are always invertible. Verify this. $]$ When $A$ and $A_{11}$ are both invertible, $(7)$ leads to a formula for $A^{-1},$ using $S^{-1}$ , $A_{11}^{-1}$ , and the other entries in $A$.

$X = A _ { 21 } A _ { 11 } ^ { - 1 }$$Y = A _ { 11 } ^ { - 1 } A _ { 12 }$

Algebra

Chapter 2

Matrix Algebra

Section 4

Partitioned Matrices

Introduction to Matrices

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

01:50

Show that the block upper …

05:37

Suppose the block matrix $…

01:38

Use matrix algcbra to show…

01:29

Let $A$ be an invertible m…

01:52

Suppose that $B$ is any ma…

02:36

Suppose $A_{11}$ is an inv…

02:51

If $A$ is an invertible $n…

10:58

Suppose $A$ is an nonsingu…

03:51

Use the equivalence of (a)…

01:20

Suppose $A$ and $B$ are $n…

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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