Watch this video lesson to learn about another method you can use to solve a matrix problem if you are given the inverse of the matrix. You will also learn the identifying mark of the multiplicative inverse of a matrix.
The Matrix Multiplicative Inverse
The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. In math symbol speak, we have A * A sup -1 = I. This tells you that when you multiply a matrix A with its multiplicative inverse, you will get the identity matrix.
Yes, we write the inverse with a superscript of -1. When we deal with regular numbers, our multiplicative inverse is the number we multiply by to get 1. So, for the number 2, it is 1/2. For the number -3, the multiplicative inverse is -1/3. For our normal numbers, the multiplicative inverse is simply 1 divided by our number.
Unfortunately, not all matrices will have an inverse, nor is finding the multiplicative inverse that simple. In order to find the multiplicative inverse, we have to find the matrix for which, when we multiply it with our matrix, we get the identity matrix. Our matrices must also be square, having the same number of rows and columns.
We will leave the discussion on how to find the inverse of a matrix to another lesson. For this lesson, we will talk about its benefits. You see, it is useful to learn about the multiplicative inverse of a matrix because if we know it, then we can use it to help us solve equations with matrices in them.
A Matrix Equation
For example, we can use it to solve a problem like this:
This matrix equation is in the form of Ax = b, where A is your coefficient matrix, x is your variable matrix, and b is your answer matrix. While we can use other methods to solve such a problem, if we know the multiplicative inverse of our coefficient matrix, then we can easily solve the problem by simply multiplying both sides by the inverse.
So, if we knew A sup -1, our answer would be x = A sup -1 * b. Yes, our answer would be our answer matrix, b, multiplied by the multiplicative inverse of our coefficient matrix. Let's look at how this works. For our coefficient matrix, we have this matrix as the multiplicative inverse matrix:
We can check whether this inverse is real or not by multiplying it with our coefficient matrix to see if we get the identity matrix. Multiplying the two matrices, we see that we do get the identity matrix:
We know for sure now that this inverse is the real inverse, and it works for us.
Using the Multiplicative Inverse
To use this inverse to help us find our answer, we simply multiply the inverse with the right side of our problem. So, we have this:
In math symbols, we have x = A sup -1 * b. We go ahead and multiply the matrices together. From the top row, we get 1(11) + -2(5) = 11 - 10 = 1. For the bottom row, we get -1/2(11) + 3/2(5) = -11/2 + 15/2 = 4 / 2 = 2. Now, we have this:
Our answer can then be easily found by just translating this back into equation form. We get x = 1 and y = 2. As you can see, using the inverse of a matrix to find our solution can be a very easy thing to do. I would use this method whenever you know what the inverse of a matrix is.
Let's review what we've learned. The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. In math symbol speak, we have A * A sup -1 = I.
You can use the multiplicative inverse of a matrix to solve problems in the form of Ax = b, where A is your coefficient matrix, x is your variable matrix, and b is your answer, or constant, matrix.
If you know the inverse of a matrix, you can solve the problem by multiplying the inverse of the matrix with the answer matrix, x = A sup -1 * b. After you multiply, you can then easily find the answer by translating back to equation form.
Following this lesson, you should be able to:
- Define multiplicative inverse of a matrix
- Explain when and how you can use the multiplicative inverse of a matrix to solve problems