Exercise 3.2.4

Prove that if A is an invertible square matrix, then A+=A1.

First we note that if A is invertable then so is its transpose:

AA1=I(AA1)T=I(A1)TAT=I

So the inverse of AT is (AT)1=(A1)T (Sometimes written as AT) 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=A1(AT)1AT=A1