Is this statement true $AA^t = A^{-1}$

Question

I want to know if its true or not.

According to what I have read this is true for orthogonal matrices. Is it true or not ?

Are there any other cases in which this could be true?

Answer 1

For an orthogonal matrix $A$ we have: $$A^{-1}=A^t,$$ which with your property gives $A=I$.

Answer 2

As you have already been told, the only orthogonal matrix with that property is the identity matrix.

Actually, there are very few matrices with that property. First of all, notice that$$A.A^t=A^{-1}\implies\det(A)^2=\frac1{\det A}$$and therefore, if we are dealing with real matrices, $\det A=1$. Furthermore, since $A.A^t$ is symmetric, then $A^{-1}$ is symmetric and so $A$ is symmetric too. So, now, in the case of $2\times 2$ matrices, we are dealing with matrices of the type $\left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ such that $ac-b^2=1$. The inverse of such a matrix is $\left(\begin{smallmatrix}c&-b\\-b&a\end{smallmatrix}\right)$. And now it is not hard to see that the only $2\times2$ real matrix with that property is the identity matrix.