# Question about inverse of matrix

• Carl Rowan
In summary, the conversation discusses how to show that the inverse of a matrix with full row rank and no terms with s or higher degree can be found when the original matrix is invertible. It is suggested to manipulate the matrix using the identity matrix and use the determinant to find entries of the inverse matrix.
Carl Rowan

## Homework Statement

[/B]
Given this matrix
##\begin{bmatrix}As+B \\ C \end{bmatrix}##
which is invertible and ##A## has full row rank. I would like to show that its inverse has no terms with ##s## or higher degree if
##\begin{bmatrix}A \\ C \end{bmatrix}##
is invertible.

## The Attempt at a Solution

The only thing I have concluded is that you can manipulate it like this:

##\begin{bmatrix}I & 0 \\ 0 & sI \end{bmatrix} \begin{bmatrix}As+B \\ C \end{bmatrix}=\begin{bmatrix}As+B \\ Cs \end{bmatrix}=s\begin{bmatrix}A \\ C \end{bmatrix}+\begin{bmatrix}B \\ 0 \end{bmatrix}##
where each ##I## is conformable with the matrix product. Not sure how to take this any further, though. I think my main problem is that I don't know how to use the rank property of ##A##.

What can you say about the determinant of ##\begin{bmatrix}As \\ C \end{bmatrix}## in terms of powers of s?
How can you use the determinant (of the original matrix) to find entries of the full inverse matrix?

Thank you for your hints, pushed me into the right direction and helped me prove the statement!

mfb

## What is the inverse of a matrix?

The inverse of a matrix is a matrix that, when multiplied by the original matrix, produces the identity matrix. In other words, it "undoes" the original matrix.

## How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use various methods such as Gaussian elimination, matrix inversion algorithms, or the adjugate formula. The method you choose will depend on the size and complexity of the matrix.

## When does a matrix have an inverse?

A matrix has an inverse if and only if it is a square matrix (same number of rows and columns) and its determinant is non-zero. If the determinant is zero, the matrix is said to be singular and does not have an inverse.

## Why is finding the inverse of a matrix important?

The inverse of a matrix is important in various mathematical applications, such as solving systems of linear equations, computing eigenvalues and eigenvectors, and performing transformations in geometry. It is also used in many machine learning and data analysis algorithms.

## Can any matrix have more than one inverse?

No, a matrix can only have one inverse. If a matrix has two inverse matrices, they must be equal. If a matrix has no inverse, it is singular and cannot have an inverse.

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