Lesson Video: Properties of Inverse Matrices Mathematics

Video Transcript

In this lesson, we will learn how to use some properties of the matrix inverse. At this point, you should be familiar with finding the determinant and the inverse of two-by-two and three-by-three matrices. Firstly, before we get started, recall the identity matrix, the matrix whose elements along the principal diagonal, that is, the diagonal from top left to bottom right, are one and the rest are zero. So the two-by-two identity matrix is one, zero, zero, one. And the three-by-three identity matrix is one, zero, zero, zero, one, zero, zero, zero, one.

Recall that a square matrix 𝐴 is called invertible if there is an 𝐴 inverse such that 𝐴 multiplied by 𝐴 inverse is the same as 𝐴 inverse multiplied by 𝐴, which gives us the identity matrix. So what we can say from this definition is that if 𝐵 is the inverse of 𝐴, then the product 𝐴𝐵 and the product 𝐵𝐴 gives us the identity matrix. And we can use this to check whether two matrices 𝐴 and 𝐵 are the inverse of each other.

So let’s now see the properties of the matrix inverse. From the definition of the matrix inverse, we have that the inverse of the inverse of 𝐴 is just 𝐴. Secondly, we have that the inverse of the product 𝐴𝐵 is the inverse of 𝐵 multiplied by the inverse of 𝐴. We’ve got to be really careful with this one because it might be easy to think that this is the inverse of 𝐴 multiplied by the inverse of 𝐵. But we know that matrix multiplication is not commutative.

We can do a quick proof to show how this works. Remember, we said that if we multiply a matrix by its inverse, we get the identity matrix. So if we take the matrix 𝐴𝐵 and multiply it by its inverse, which is 𝐵 inverse multiplied by 𝐴 inverse, we should get the identity matrix. So let’s double-check this. Because of the associative property of matrix multiplication, this is the same as 𝐴 multiplied by 𝐵 multiplied by 𝐵 inverse multiplied by 𝐴 inverse. And we know that 𝐵 multiplied by 𝐵 inverse must give us the identity matrix, because that’s just from the definition of an invertible matrix. So we have 𝐴 multiplied by the identity matrix multiplied by 𝐴 inverse. But multiplying any matrix by the identity matrix just gives us the same matrix. So this is just 𝐴 multiplied by 𝐴 inverse. And again from the definition of an invertible matrix, we know that 𝐴 multiplied by 𝐴 inverse just gives us the identity matrix.

So because when we multiply 𝐴𝐵 by its inverse, which is 𝐵 inverse multiplied by 𝐴 inverse, and we get the identity matrix, we’ve shown that 𝐵 inverse multiplied by 𝐴 inverse is definitely the inverse of 𝐴𝐵. And for a conclusive proof, we could show by the same method that 𝐵 inverse multiplied by 𝐴 inverse multiplied by 𝐴𝐵 also gives us the identity matrix.

Let’s move on and look at the third property of the matrix inverse. 𝐴 transposed inverse is equal to 𝐴 inverse transposed. Remember, this notation 𝑇 means the matrix transpose. And we transpose a matrix by swapping the rows with the columns. As an example, if we have the matrix 𝐴 one, four, six, two, then the matrix 𝐴 transposed is equal to one, six, four, two. Again, we can check this result because we know that we can take a matrix and multiply it by its inverse to get the identity matrix. So if we take the matrix 𝐴 transposed and multiply it by its inverse, which is 𝐴 inverse transposed, we should get the identity matrix.

To proceed from here, we recall the property of the matrix transpose. The transpose of 𝐴𝐵 is 𝐵 transpose multiplied by 𝐴 transpose. That means this is equal to 𝐴 inverse multiplied by 𝐴 transposed. But from the definition of the matrix inverse, we know that 𝐴 inverse multiplied by 𝐴 gives us the identity matrix. So we get the identity matrix transpose. But if we transpose the identity matrix, we just get the identity matrix. So we’ve shown that multiplying 𝐴 transpose by 𝐴 inverse transpose gives us the identity matrix, which confirms this property. And for a conclusive proof by a similar method, we can show that 𝐴 inverse transpose multiplied by 𝐴 transpose also gives us the identity matrix.

And finally we have that 𝐴 to the 𝑛th power inverse is equal to 𝐴 inverse to the 𝑛th power. Again, we can show this by proving that 𝐴 to the 𝑛th power multiplied by 𝐴 inverse to the 𝑛th power is going to give us the identity matrix. We can see that we’re going to be able to pair off the 𝐴’s with the 𝐴 inverse. And each time we do this, we’ll get the identity matrix. We know that we have the same number of 𝐴’s as 𝐴 inverses. So by pairing these off and multiplying them together, we know we’re going to end up with the identity matrix. And by a similar method, we can show that the inverse of 𝐴 to the 𝑛th power multiplied by 𝐴 to the 𝑛th power also gives us the identity matrix. So let’s now see how we can use these properties to answer some questions.

If 𝐴 is a matrix, which of the following is equal to 𝐴 inverse squared? 𝐴 to the half power, 𝐴 squared, 𝐴 inverse to the half power, or 𝐴 squared inverse.

We can answer this question by recalling the property of inverse matrices. That is, 𝐴 to the 𝑛th power inverse equals 𝐴 inverse to the 𝑛th power, for 𝑛 is a positive integer. So with this in mind, we can say that 𝐴 inverse squared is equal to 𝐴 squared inverse. But let’s double-check this relation just to be sure. We found that the inverse of 𝐴 squared is 𝐴 inverse squared. And we know if we take a matrix and multiply it by its inverse, we should get the identity matrix. So let’s check this.

If we take the matrix 𝐴 squared and multiply it by its inverse 𝐴 inverse squared, we should get the identity matrix. We can just write this as 𝐴 multiplied by 𝐴 multiplied by 𝐴 inverse multiplied by 𝐴 inverse. And because of the associativity property of matrix multiplication, we can write it in this way. We know 𝐴 multiplied by 𝐴 inverse gives us the identity matrix. And we know multiplying any matrix by the identity matrix just gives us the same matrix. So this is just 𝐴 multiplied by 𝐴 inverse, which gives us the identity matrix.

If 𝐴 is a matrix, which of the following is equal to 𝐴 inverse transpose?

Recall that this notation means the transpose of a matrix. This just means we switch the rows with the columns. For example, if we have the matrix 𝑋 equals two, one, six, seven, 𝑋 transpose is equal to two, six, one, seven. To answer this question, we recall the property of the matrix inverse. That is, 𝐴 transposed inverse is equal to 𝐴 inverse transposed. And this gives us that our answer is the first option. 𝐴 inverse transposed is equal to 𝐴 transposed inverse. So what we’re saying is that if we take the matrix 𝐴, find its inverse, and then transpose it, this is exactly the same result we would get by taking the matrix 𝐴, transposing it, and then finding the inverse.

Consider the matrix 𝐴 equals negative three, one, negative two, five. Find 𝐴 inverse inverse.

Recall the definition of the inverse of 𝐴 is the matrix such that 𝐴 multiplied by 𝐴 inverse equals the identity matrix. We do have a method for finding the inverse of a two-by-two matrix. That is, given the matrix 𝑋 equals 𝑎, 𝑏, 𝑐, 𝑑, 𝑋 inverse is equal to one over 𝑎𝑑 minus 𝑏𝑐 multiplied by the matrix 𝑑, negative 𝑏, negative 𝑐, 𝑎. So to find the inverse of the inverse of 𝐴, we could use this method to find the inverse of 𝐴 and then repeat the method to find the inverse of that.

However, we do have one property of the matrix inverse that can help us to do this a little bit quicker. That is, the inverse of the inverse of 𝐴 is just equal to 𝐴. So if we take a matrix and find its inverse and then invert that matrix, we get the original matrix back. So, actually, the inverse of the inverse of our matrix 𝐴 is just the matrix 𝐴 negative three, one, negative two, five.

Given the matrices 𝐴 and 𝐵, where 𝐴 equals one, negative two, three, zero, negative one, four, zero, zero, one and 𝐵 equals one, negative two, five, zero, negative one, four, zero, zero, one, find 𝐴𝐵. And the second part of the question says, “Without doing any further calculations, find 𝐴 inverse.”

So the first thing we’re going to do here is find the product 𝐴𝐵. Using the usual method for multiplying three-by-three matrices together, we find that 𝐴𝐵 is one, zero, zero, zero, one, zero, zero, zero, one. And we notice that this is actually the three-by-three identity matrix. So what does this mean for our matrices 𝐴 and 𝐵?

Well, the definition of the inverse matrix is that it’s the 𝐴 inverse such that 𝐴 multiplied by 𝐴 inverse equals the identity matrix. So the fact that we found the product 𝐴𝐵 to be the identity matrix means that the matrix 𝐵 must be the inverse of the matrix 𝐴.

The second part of the question says, “Without doing any further calculations, find 𝐴 inverse.” Well, because when we find the product 𝐴𝐵 we get the identity matrix, this means that the matrix 𝐵 is the inverse of 𝐴. Therefore, 𝐴 inverse is the matrix 𝐵, which is one, negative two, five, zero, negative one, four, zero, zero, one.

Given that 𝐴𝐵 inverse equals one-sixth multiplied by five, negative three, negative 33, 21 and 𝐴 equals negative two, negative one, negative three, negative two, determine 𝐵 inverse.

Let’s start with a quick reminder of what the matrix inverse is. The inverse of a square matrix 𝐴, 𝐴 inverse, is the matrix such that 𝐴 multiplied by 𝐴 inverse gives us the identity matrix. And one property of the matrix inverse which is going to prove useful to us here is 𝐴𝐵 inverse is equal to 𝐵 inverse multiplied by 𝐴 inverse. So because we’re told the inverse of the product 𝐴𝐵 is one-sixth multiplied by the matrix five, negative three, negative 33, 21, then from this property of the matrix inverse, we can say that this is the same as 𝐵 inverse multiplied by 𝐴 inverse. But how is this going to help us find 𝐵 inverse?

Well, there’s a little bit of a trick that we can apply here. And all it requires is remembering the definition of the matrix inverse. We can find the matrix 𝐵 inverse, 𝐴 inverse, 𝐴 by multiplying 𝐵 inverse, 𝐴 inverse by 𝐴 on the right. So let’s now multiply these two matrices together to see what we get. We do this in the usual way of multiplying two two-by-two matrices together. And then we can simplify each entry. And we end up with one-sixth multiplied by the matrix negative one, one, three, negative nine. We then remember when we have a scalar multiplied by a matrix, we can just multiply each entry by that scalar. And that gives us the matrix negative one over six, one over six, one over two, negative three over two. But how has this actually helped us find the matrix 𝐵 inverse?

Well, what we found is the matrix 𝐵 inverse multiplied by 𝐴 inverse multiplied by 𝐴. And from the definition of the matrix inverse, 𝐴 inverse multiplied by 𝐴 gives us the identity matrix. So what we’ve actually found is the matrix 𝐵 inverse multiplied by the identity matrix. But multiplying any matrix by the identity matrix just gives us that matrix. So what we’ve found is the matrix 𝐵 inverse. So by using the definition of the matrix inverse and one of the properties of the matrix inverse, we were able to find an unknown using the matrix inverse.

Let’s now summarize the key points from this lesson. For square matrices 𝐴 and 𝐵, we have that the inverse of the inverse of 𝐴 is just equal to 𝐴. The inverse of the product 𝐴𝐵 is equal to 𝐵 inverse multiplied by 𝐴 inverse. 𝐴 transpose inverse is equal to 𝐴 inverse transpose. So whether we transpose the matrix and then invert it or invert the matrix and then transpose it, we get the same result. And finally, raising the matrix 𝐴 to the 𝑛th power and then inverting it gives us the same result as inverting the matrix 𝐴 and then raising it to the 𝑛th power. And that’s for a positive integer 𝑛.