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Amy has a master's degree in secondary education and has taught math at a public charter high school.
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:
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.
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.
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.
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:
So we first write this matrix next to its matching sized identity matrix.
We get one big matrix.
Now we can use matrix operations to turn the first half of this matrix into an identity matrix. What we want to accomplish is to turn all the numbers not on the diagonal to zeros and all the numbers on the diagonal to ones. Your matrix operations may be in a different order than mine, but the end result will always be the same if done correctly.
What I'm going to do is first add the second row to the third row to get a new third row. I get 0, 0, 1, 0, 1, 1. Now I'm going to multiply this new third row by -3 and add it to the top row to get a new top row. Multiplying the third row by -3, I get 0, 0, -3, 0, -3, -3. Adding this to the top row, I get 1, 0, 0, 1, -3, -3 for my new top row. My last step to turn the left side of this matrix into an identity matrix is to divide the second row by 2. Doing this, I get 0, 1, 0, 0, 0.5, 0 for my new second row. Now that I've turned the left side of the matrix into an identity matrix, the right side gives my inverse matrix. My inverse matrix is this one.
And I'm done. I have found my inverse matrix.
Let's review what we've learned now. We've learned that the identity matrix is the square matrix that has ones on its diagonal and zeros everywhere else, and an 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. We can liken our identity matrix as the number 1 of the matrix world. Just like how anything multiplied by 1 is itself, so it is with the identity matrix. A matrix multiplied by its matching identity matrix is itself.
To find the inverse of a matrix we set up a big matrix by combining our matrix with its identity matrix. We write our matrix on the left and the identity matrix on the right. We perform matrix operations to turn the left side into the identity matrix. The resulting right side will be our inverse matrix. Inverse matrices are important to learn about because they act as division in the matrix world. By using an inverse matrix, we can solve equations that involve matrices.
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Back To CourseAlgebra II Textbook
26 chapters | 256 lessons