If #A^T# is invertible, is A invertible? What about #A^TA#?

1 Answer
Nov 7, 2015

Answer:

Yes and yes

Explanation:

Suppose #A^T# has inverse #(A^T)^(-1)#

For any square matrices #A# and #B#, #A^T B^T = (BA)^T#

Then:

#((A^T)^(-1))^T A = ((A^T)^(-1))^T (A^T)^T=(A^T (A^T)^(-1))^T = I^T = I#

And:

#A ((A^T)^(-1))^T = (A^T)^T ((A^T)^(-1))^T=((A^T)^(-1) A^T)^T = I^T = I#

So #((A^T)^(-1))^T# satisfies the definition of an inverse of #A#.

Then we find:

#(A^T A) (A^-1 (A^T)^(-1)) = A^T (A A^-1) (A^T)^(-1)#

#=A^T I (A^T)^(-1) = A^T (A^T)^(-1) = I#

And:

#(A^-1 (A^T)^(-1)) (A^T A) = A^(-1)((A^T)^(-1) A^T)A#

#=A^(-1) I A = A^(-1)A = I#

So #(A^-1 (A^T)^(-1))# satisfies the definition of an inverse of #A^T A#