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.