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ACT Prep: Tutoring Solution43 chapters | 384 lessons

Instructor:
*Sharon Linde*

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.

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.

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:

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:

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:

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.

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:

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:

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.

Now that we have the determinant of our matrix, all we have to do is find the adjugate matrix. The **adjugate matrix** is found by first finding the transpose matrix, finding the cofactors of the transpose matrix, and finally applying alternating signs to the transpose matrix. The **transpose matrix** is just our same matrix but with the rows turned into columns and columns turned into rows. Some people like to think of this as flipping the matrix about the main diagonal. Either way you look at it is fine, because either way you'll end up with this:

The next step is to replace each element in this transposed matrix with its cofactor, using the same process we just went over for finding the determinants of the submatrix.

After finding the determinants of all nine of these 2x2 matrices, we end up with the following our matrix:

Now that we have all those values, we need to change some of the signs in this matrix. We start with a positive in the upper left corner and alternate with negatives, like this.

Applying these sign changes leads to our final adjugate matrix of:

Now that we have our adjugate matrix, all we need to do is multiply by 1 over the determinant. In our matrix, our determinate is 2, so this means we multiply the adjugate matrix by ½, which results in our final inverse matrix:

Whew! After all of that work, our final answer for the inverse matrix is:

Now that we have our final inverse matrix, how do we know it is the correct answer? Well, if we go back to our original inverse matrix equation . . .

we can see all we have to do is multiply our original matrix by our answer and we should get the identity matrix. Let's see how we do:

And there we have it! Since our answer solves the original inverse matrix equation, we know we have done our work correctly. If we had gotten something other than the identity matrix, then we would know that we had made a mistake somewhere along the way.

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ACT Prep: Tutoring Solution43 chapters | 384 lessons

- What is a Matrix? 5:39
- How to Write an Augmented Matrix for a Linear System 4:21
- How to Perform Matrix Row Operations 5:08
- Matrix Notation, Equal Matrices & Math Operations with Matrices 6:52
- How to Solve Inverse Matrices 6:29
- Finding the Inverse of a 3x3 Matrix
- How to Solve Linear Systems Using Gaussian Elimination 6:10
- How to Solve Linear Systems Using Gauss-Jordan Elimination 5:00
- Inconsistent and Dependent Systems: Using Gaussian Elimination 6:43
- Multiplicative Inverses of Matrices and Matrix Equations 4:31
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- Finding the Determinant of a 3x3 Matrix
- How to Find the Determinant of a 4x4 Matrix 5:41
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