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How to Find the Inverse of a 4x4 Matrix

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  • 0:00 Steps to Solving the Problem
  • 4:25 Checking Your Work
  • 5:26 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson defines a matrix and some related terms, as well as outlining the rules and guidelines for working with matrices. Once these terms are defined, we will discuss how to find the inverse of a 4x4 matrix.

Steps to Solving the Problem

Before we get to the steps of finding the inverse of a 4x4 matrix, let's do a quick review of some definitions and properties. A matrix is an array of numbers. An a x b matrix has a rows and b columns. We are working with a 4x4 matrix, so it has 4 rows and 4 columns. An example of a 4x4 matrix R is shown in the following image.

4x4 Matrix
4x4-1

An n x n matrix is a matrix with an equal number of rows and columns. The identity matrix is an n x n matrix that is represented with the letter I. This matrix has ones along the diagonal and zeros everywhere else. Think of the identity matrix as the number 1. If you multiply any number by 1, you'll get that original number back. Just the same, for any n x n matrix A, multiplying A by the identity matrix I will give you A. In other words, AI = IA = A.

The inverse of a matrix A is the matrix B, such that AB = BA = I. Not all matrices have an inverse, but if a matrix does have an inverse, then this is the property it follows. That is, if we multiply two matrices together both ways, then we get the identity matrix in both instances.

To multiply two matrices together, find the dot product of the rows and columns of the two matrices by multiplying matching numbers and then summing them up. For instance, to find the entry in the first row and the first column of AB, take the dot product of the first row of matrix A and the first column of matrix B. Multiply each corresponding entry together, and then add up all the products. This number will be equal to the entry in the first row and the first column of the matrix AB. In general, the entry in the ith row and jth column of AB will be equal to the dot product of the ith row of A and the jth column of B. The following image shows multiplication of 2x2 matrices, but the pattern holds for multiplying any two n x n matrices together.

Multiplying 2x2 Matrices
4x4-3

The last thing to review before finding the inverse of a 4x4 matrix is row operations. There are three row operations that we can perform on a matrix to produce an equivalent matrix. Row switching allows the interchange of any two rows. Row multiplication permits any row to be multiplied by a scalar. (Note that when we are working with matrices, we call regular numbers scalars so we don't confuse them with a 1x1 matrix.) The last row operation is row addition, which allows us to add any two rows together and then replace one of those rows with the result.

Now that we've gotten the basics out of the way, let's talk about how to find the inverse of a 4x4 matrix. In general, there are three basic steps when finding the inverse of an n x n matrix A. These steps hold true for a 4x4 matrix.

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