Matrix Inversion: Proving M is Invertible

In summary: It's not hard to see what will happen when you multiply \begin{bmatrix}A & C \\ 0 & B\end{bmatrix} with \begin{bmatrix}A^{-1} & X \\ 0 & B^{-1}\end{bmatrix}.In summary, by using block multiplication and the property that a matrix is invertible if its determinant is nonzero, we can see that the matrix M composed of A, C, O, and B is invertible if and only if all entries of C are equal to zero. This can be proven by rearranging equations and using the fact that any matrix times a zero matrix is equal to zero.
  • #1
Frillth
80
0

Homework Statement



Suppose A is an invertible mxm matrix, B is an invertible nxn matrix, and C is an arbitrary mxn matrix. Is the matrix M =

A|C
----
O|B

invertible? Solve with proof.

Hint: Use block multiplication.

Note: I'm not quite sure how to draw this matrix on the forums. It should me an (m+n)x(m+n) matrix with the entries of A in the top left, the entries of C in the top right, zeros in the bottom left, and the entries of B in the bottom right. I hope that's clear.

Also, O is a matrix with all entries equal to zero.

Homework Equations



For some mxm matrix M:

M*M^-1 = M^-1*M = I_m

The Attempt at a Solution



I showed in a different problem that a matrix composed like in this problem, but with C being all zeros is invertible, so my approach for this problem is to assume that the matrix has an inverse, and then show that all of C's entries must be 0.

Let the matrix's inverse M^-1 be:
A`|C`
------
O`|B`
With A`, C`, O`, and B` being some matrices with the same dimensions as their non primed counterparts. Now multiply M^-1*M =
A`A + C`O|A`C + C`B
----------------------
O`A + B`O|O`C + C`B
Then multiply M*M^-1:
AA` + CO`|AC` + CB`
----------------------
OA` + BO`|OC` + BB`
This must be equal to the (m+n)x(m+n) identity matrix, which can be written as:
I_m| O
--------
O |I_n

With this knowledge and the fact that any matrix times O = O, we can write the following equations:

nxm matrices:
O`A + B`O = OA` + BO` = O
mxn matrices:
A`C + C`B = AC` + CB` = O
mxm matrices:
CO` = O
nxn matrices:
O`C = O

I also know:
A*A` = A`*A = I_m
B*B` = B`*B = I_n

I've tried rearranging these a little bit, but nothing has come of it so far. Am I on the right track? If so, could somebody nudge me in the right direction? If not, how should I attack this problem?

Thanks!

Edit: I just realized that I can do the following
O`A = O
Multiply both sides by A` = A^-1
O`AA` = OA`
O`I_m = O
O` = O
This doesn't really help me prove that C is all zeros though.
 
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  • #2
If you just multiply by (A^(-1)|B^(-1)), or however you write that, don't you get an upper triangular matrix with one's down the diagonal? Isn't the determinant of that nonzero? In fact, isn't it one?
 
  • #3
Wait, how do determinants come into play here?
 
  • #4
A matrix is invertible if it's determinant is nonzero. Isn't it?
 
  • #5
I actually didn't know that. Thank you.

However, I'm not sure that I'm supposed to use that method. I don't recall proving that in class, and we haven't learned how to find determinants of matrices larger than 3x3.

Is there another way that I could solve this?
 
  • #6
If you have an upper triangular matrix with ones down the diagonal, it's also pretty easy to see what row operations would reduce it to the identity. Is that proof enough?
 
  • #7
You can also pretty much write down the inverse of that matrix. It's easy to see that it has to be of the form

[tex]\begin{bmatrix}A^{-1} & X \\ 0 & B^{-1}\end{bmatrix}.[/tex]

You only have to solve for X.
 

1. What is matrix inversion?

Matrix inversion is the process of finding the inverse of a square matrix, denoted as M-1. The inverse of a matrix is another matrix that, when multiplied with the original matrix, results in the identity matrix. In other words, it is a way to "undo" the effects of a matrix multiplication.

2. Why is it important to prove that M is invertible?

It is important to prove that M is invertible because it ensures that the matrix has a unique solution for every system of linear equations. This means that the matrix is not singular (has a determinant of 0) and can be used in various calculations and transformations.

3. How do you prove that a matrix is invertible?

There are a few different methods for proving that a matrix is invertible, but the most common is by using Gaussian elimination to reduce the matrix to its reduced row echelon form. If the reduced matrix has a pivot in every column, then the original matrix is invertible. Another method is by calculating the determinant of the matrix - if it is non-zero, then the matrix is invertible.

4. Can any square matrix be inverted?

No, not all square matrices can be inverted. A matrix must meet certain conditions in order to be invertible, such as having a non-zero determinant and a unique solution for every system of linear equations. If a matrix does not meet these conditions, it is called singular and cannot be inverted.

5. What are some real-world applications of matrix inversion?

Matrix inversion has many applications in fields such as physics, engineering, and computer science. It can be used to solve systems of linear equations, transform geometric shapes, and calculate the inverse of a function. In machine learning and data analysis, matrix inversion is used for tasks such as dimensionality reduction and solving optimization problems.

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