Finding the Inverse of a 3x3 Matrix

Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

Do you know what the inverse of a 3x3 matrix is and how to find it? This lesson goes over these and related concepts necessary for finding the inverse of a sample 3x3 matrix.

Setting up the Problem

Why would you ever need to find the inverse of a 3x3 matrix? Well, matrices and inverse matrices have lots of applications in geometry, the sciences, and especially computer science. Let's say you are a computer designer, or want to be one someday, and you need to take the matrix below and find its inverse.


3x3 Inverse


To find the inverse of a 3x3 matrix, we first have to know what an inverse is. Mathematically, this definition is pretty simple. Just check out the equation below:


3x3 Inverse


Let's look at all the terms in the above matrix equation see what they mean in plain English. M is just our original matrix. M raised to the power of -1 is the mathematical symbol for the inverse matrix of M. And finally, I is the identity matrix, which has 1s on the main diagonal and 0s everywhere else. It looks like this:


3x3 Inverse


So, the inverse matrix is the matrix we'll have to multiply our original matrix by to get the identity matrix. Simple enough in concept, right? But how do you calculate a matrix inverse? For that, we'll turn to another matrix equation:


3x3 Inverse


This equation shows us that to find the inverse matrix we have to find the adjugate matrix and divide by the determinant. Let's look at these one at a time, starting with the determinant.

Determinant

The determinant of a matrix is a single number that is characteristic of that matrix. You can find the determinant using several methods. For this exercise, we'll use a method that uses expansion along a row or column. Why? Because this method reduces the number of calculations if you have any zeros in your matrix. Since we have both a zero in our matrix and would like to reduce the number of calculations, this method will work well for us.

To use the expansion method, we'll need to multiply each term in the row or column we choose by its cofactor. Cofactors are the determinants of the submatrix of a matrix element that does not include the rows or column of that element. This sounds confusing, but it's really pretty simple.

For example, the cofactor of the matrix element of M in the first row and first column will be the determinant of the submatrix that does not include any elements from either the first row (1, 2, 3) or first column (1, 0, 1). The first row, first column element expansion times its cofactor looks like this:


3x3 Inverse


Continuing our expansion along the first column, we will have the first column (1, 0, 1), second row (0, 1, -2) element times the cofactor. But wait! That element is equal to 0, and anything multiplied by 0 is just 0. We don't have to calculate anything for this term. This saves us a step, which is what we wanted in the first place.

The last term in our expansion is going to be the first column (1, 0, 1), third row (1, 2, 5) and will look like this:


3x3 Inverse


When we add all of our expansion elements up we get 9 + 0 - 7 = 2. This means the determinant of our matrix is equal to 2. A lot of work for a small number, but we are making progress.

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