How to Solve Inverse Matrices

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  • 0:02 The Identity Matrix
  • 1:10 The Inverse Matrix
  • 2:42 Finding the Inverse Matrix
  • 3:27 Using Matrix Operations
  • 5:04 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn what kinds of matrix operations you can take to find the inverse of a matrix. Also learn why matrix inverses are useful.

The Identity Matrix

In this video lesson, we will talk about inverse matrices, but before we can introduce inverse matrices, we need to know about the identity matrix. The identity matrix is the square matrix that has ones on its diagonal and zeros everywhere else. Think of the identity matrix as the number 1 in the matrix world. These are all examples of identity matrices:

Identity Matrices
inverse matrix

Do you see how each of these identity matrices are all square, meaning that they have the same number of rows and columns? Also, all the numbers are 0 except the numbers making up the diagonal, which are all 1s. The diagonal is the line that starts at the upper left of the matrix and ends at the bottom right.

These are called identity matrices because matrices that are multiplied by their matching sized identity matrix will produce the original matrix. We label an identity matrix with a capital I. So, matrix A multiplied by its identity matrix I will equal matrix A.

The Inverse Matrix

Now that we've covered the identity matrix, we can now talk about the inverse matrix. We label inverse matrices with a superscript of -1. So the inverse matrix is defined as the inverse matrix that meets the criteria of A * A sup -1 = I, where A stands for a matrix A, A sup -1 stands for the inverse of matrix A, and I stands for the identity matrix.

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Yes, if we multiply a matrix by its inverse, then we will get the identity matrix as our answer. This is just like when we multiply a whole number by its inverse, we get the number 1. For example, 1/9 is the inverse of 9. When we multiply them together, we get 1. We can also write 1/9 as 9 sup -1.

Inverse matrices are important in the matrix world because we can't divide in the matrix world. But by using an inverse matrix, we are essentially dividing. To link this to the real world, think of the the inverse of the number 9, 1/9. Aren't we dividing by 9?

One other important thing to note about inverse matrices is that not all matrices will have an inverse matrix. This is just the nature of the matrix world. Just like we can't divide matrices, we can't always find an inverse matrix.

Finding the Inverse Matrix

To find the inverse of a particular matrix, we are going to write our matrix and its matching sized identity matrix right next to each other in one big matrix. [A | I]. Then, we are going to use matrix operations to change the first matrix into the identity matrix. What used to be the identity matrix on the right side will now be the inverse matrix. [I | A sup -1]. It is like mathematical magic! It just works! Giving a proof of this method, though, is beyond the scope of this lesson. Let's see how this is done with an example, then.

We will try to find the inverse matrix of this matrix:

inverse matrix

So we first write this matrix next to its matching sized identity matrix.

inverse matrix

We get one big matrix.

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