If 𝐴 and 𝐵 are both nonsingular matrices, then what is the value of the inverse of 𝐴 times 𝐵?
In this question, we’re given two nonsingular matrices 𝐴 and 𝐵. And we need to determine the value of the inverse of their product. To answer this question, let’s start by recalling what it means to say that a matrix is nonsingular. This means that the matrix is invertible. In other words, we know that 𝐴 inverse and 𝐵 inverse both exist. And there’s one other useful property about matrices 𝐴 and 𝐵 this tells us. It tells us that they must be square matrices because only square matrices are invertible. And this is almost enough to help us find an expression for the inverse of matrix 𝐴 multiplied by 𝐵.
However, there is one thing that’s assumed here that we can multiply matrices 𝐴 and 𝐵 together. And of course since matrices 𝐴 and 𝐵 are both square matrices, this just means they have the same order.
We’re now ready to recall the following fact about the properties of invertible matrices. If we have two nonsingular matrices 𝐴 and 𝐵 of the same order, the inverse of 𝐴 times 𝐵 is 𝐵 inverse multiplied by 𝐴 inverse. And this is of course enough to just answer our question.
However, we can also ask a new question. Why does this property hold true? And we can do this by just finding an inverse of matrix 𝐴 multiplied by 𝐵. And there’s many ways of doing this. Let’s call the matrix 𝐴 multiplied by 𝐵 matrix 𝐶. So 𝐶 is equal to 𝐴 times 𝐵. And remember, 𝐴 and 𝐵 are both invertible matrices. So they’re both square matrices of the same order. That means 𝐶 is also a square matrix of the same order. And finding the inverse of 𝐴 multiplied by 𝐵 means we’re trying to find the inverse of 𝐶. That’s the matrix which satisfies the equation 𝐶 times 𝐶 inverse is equal to the identity matrix. And 𝐶 is the matrix 𝐴𝐵. So we can rewrite this equation as 𝐴𝐵 times the inverse of 𝐴𝐵 is equal to the identity matrix.
We can then solve this for the inverse of 𝐴𝐵. 𝐴 is an invertible matrix. So we can multiply on the left-hand side of this equation by the inverse of 𝐴. This gives us 𝐴 inverse times 𝐴 times 𝐵 multiplied by the inverse of 𝐴𝐵 is equal to 𝐴 inverse times the identity matrix. And of course we can simplify this. 𝐴 inverse times 𝐴 is the identity matrix. And multiplying by the identity matrix doesn’t change the value. So this equation simplifies to give us 𝐵 times the inverse of 𝐴𝐵 is equal to 𝐴 inverse.
We can do this one more time. 𝐵 is also an invertible matrix. So we can multiply on the left by the inverse of 𝐵. This then gives us 𝐵 inverse times 𝐵 multiplied by the inverse of 𝐴𝐵 is equal to 𝐵 inverse times 𝐴 inverse. And once again we simplify. 𝐵 inverse multiplied by 𝐵 is the identity matrix. And this then leaves us with our result, which we can use to answer our question.
For nonsingular matrices 𝐴 and 𝐵 of the same order, the inverse of 𝐴 times 𝐵 is equal to 𝐵 inverse multiplied by 𝐴 inverse.