The inverse of an orthogonal matrix is its transpose
An orthogonal matrix has orthogal columns, i.e. the scalar product of two different columns is zero (the case $i\neq j$). For the case $i=j$ you have $a_i^Ta_i=\|a_i\|^2&gt;0$. 

So, all you can say about an orthogonal matrix with colums $a_i$ is:

$$
a_i^Ta_j=\begin{cases}c_i&gt;0 &amp; i=j\\ 0 &amp; i\neq j\end{cases}$$

where $c_i&gt;0$ is some constant, depending on the matrix. 

Then, there are orthonormal matrices. These fulfill, in addition to the previous statement, $c_i=1$ and hence $A^TA=I$.

Finally, only if your $A$ is a square orthonormal matrix, it holds $A^{-1}=A^T$.

The task you describe has two flaws:


it only talks about ortogonal matrices, where we can just say $A^TA=\text{diag}(\vec{c})$ with $\vec{c}$ contains the norms of the columns.
It states $A$ is an $n\times m$ matrix (i.e. not square). Rectangular matrices dont have an inverse. So, the last statement is not correct.


Edit: Normally in literature, orthogonal matrices are considered to be square with unit-norm columns. Then $A^TA=I$ and $A^{-1}=A^T$. However, as the question explicitely mentions rectangular matrices, I want to point out that it really depends on your definition of what an orthogonal matrix is. And, if you just define it as orthogonal columns, the arguments above hold.