# Question Video: Understanding the Inverse of an Inverse Matrix Mathematics

Consider the matrix 𝐴 = [−3, 1 and −2, 5]. Find (𝐴⁻¹)⁻¹.

01:10

### Video Transcript

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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.