Exercise 3.2.4
Prove that if A is an invertible square matrix, then A+=A−1.
First we note that if A is invertable then so is its transpose:
AA−1=I(AA−1)T=I(A−1)TAT=I
So the inverse of AT is (AT)−1=(A−1)T (Sometimes written as A−T) So with that we can use the fact that the inverse of a product of two matrices is the product of the inverses in reverse order to find:
A+=(ATA)−1AT=A−1(AT)−1AT=A−1