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

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Answer 1 · 2015-11-07T10:51:54

Yes and yes

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 $A^T A$

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