Question Video: Understanding the Inverse of an Inverse Matrix Mathematics

Consider the matrix ๐ด = [โˆ’3, 1 and โˆ’2, 5]. Find (๐ดโปยน)โปยน.

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Video Transcript

Consider the matrix ๐ด equals negative three, one, negative two, five. Find ๐ด inverse inverse.

Recall the definition of the inverse of ๐ด is the matrix such that ๐ด multiplied by ๐ด 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 ๐‘‹ equals ๐‘Ž, ๐‘, ๐‘, ๐‘‘, ๐‘‹ inverse is equal to one over ๐‘Ž๐‘‘ minus ๐‘๐‘ multiplied by the matrix ๐‘‘, negative ๐‘, negative ๐‘, ๐‘Ž. So to find the inverse of the inverse of ๐ด, we could use this method to find the inverse of ๐ด 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 ๐ด is just equal to ๐ด. 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 ๐ด is just the matrix ๐ด negative three, one, negative two, five.

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