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Question Video: Finding an Unknown Matrix Using the Inverse of a Matrix Mathematics

Given that (𝐴𝐡)⁻¹ = (1/6)[5, βˆ’3 and βˆ’33, 21], 𝐴 = [βˆ’2, βˆ’1 and βˆ’3, βˆ’2] determine 𝐡⁻¹.

02:25

Video Transcript

Given that 𝐴𝐡 inverse equals one-sixth multiplied by five, negative three, negative 33, 21 and 𝐴 equals negative two, negative one, negative three, negative two, determine 𝐡 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 𝐴𝐡 inverse is equal to 𝐡 inverse multiplied by 𝐴 inverse. So because we’re told the inverse of the product 𝐴𝐡 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 𝐡 inverse multiplied by 𝐴 inverse. But how is this going to help us find 𝐡 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 𝐡 inverse, 𝐴 inverse, 𝐴 by multiplying 𝐡 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 𝐡 inverse?

Well, what we found is the matrix 𝐡 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 𝐡 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 𝐡 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.

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