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### Course: Precalculus>Unit 7

Lesson 13: Introduction to matrix inverses

# Inverse matrix introduction

The inverse of a square matrix is another matrix (of the same dimensions), where the multiplication (or composition) of the two matrices results in the identity matrix. This is analogous to inverse functions (if we think of matrices as functions) or reciprocal numbers (if we think of matrices as special numbers). Fascinating! Created by Sal Khan.

## Want to join the conversation?

• Sal claims that f(f^-1(x)) = x but I'm not sure I understand the reasoning.
Any help would be appreciated! :)

I did take an example (f(x) = 2x) and saw that this is indeed, true.
However, I don't "see" why this works.
• Let f and f^-1 be inverse functions. Now, suppose f^-1(x) = y. This should mean that f(y) = x because f is the inverse of f^-1. If we substitute f^-1(x) for y, we get f(f^-1(x)) = x.
• With regular functions, we compose a function and its inverse to get x. f^-1(f(x)) = x.

How come, with matrices, we multiply a matrix and its inverse? Why don't we compose a matrix and its inverse?
(1 vote)
• what is b means in this equation
(1 vote)
• Sal has a way of making square brackets that looks like the upper left angle has a minus sign.
or it's just me?
(1 vote)
• What would be the inverse of a zero matrix?
(1 vote)
• The inverse of a zero matrix does not exist
(1 vote)
• How can I find the inverse of a 3*3 matrix?

- Aashay Joglekar
• To find the inverse of a 3x3 matrix, you can use the following steps:

Write down the 3x3 matrix you want to invert and label it as A.
Write down the identity matrix of the same size as A, and label it as I.
For example, if A is a 3x3 matrix, then I would be a 3x3 matrix with 1's on the diagonal and 0's everywhere else.
Combine A and I to form an augmented matrix [A|I].
Use elementary row operations to transform [A|I] into the reduced row echelon form [I|A^-1].
The left half of the matrix should be the identity matrix, and the right half should be the inverse of A.
If you can't get [A|I] to reduce to [I|A^-1], then the matrix A is not invertible.
It's important to note that finding the inverse of a matrix is not always possible. A matrix is invertible if and only if its determinant is nonzero. If the determinant is zero, then the matrix is said to be singular and does not have an inverse.