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,