Determine the dimension and find the actual matrix X that satisfies the matrix equation:
$$(XA)^T=(A^T+3I)X^T,$$
where A is a 3x3 matrix and $T$ denotes transpose.
My answer:
On the left side, we have nx3 matrix multiplied by 3x3 matrix, so the result is nx3, taking transpose we have 3xn. On the right side, we have 3x3 matrix (in brackets) times 3xn matrix, so the result is 3xn. The same as above. So my answer is that X is nx3, where n is arbitrary.
I would be grateful if you could confirm whether my reasoning is correct. What should I do next?
Edit:
We use the fact that $(XA)^T=A^TX^T$, therefore
$$A^TX^T=(A^T+3I)X^T$$
so
$$A^T = A^T+3I$$
which leads to
$$0=3I$$
which is a contradiction, so the answer is that there is no matrix $X$ which satisfies that equation.
$()^=(^+3)^= A^T X^T+3X^T $
By definition of transpose: $()^= A^TX^T$, hence $ A^TX^T= A^T X^T+3X^T $
thus $ 3X^T = 0 \Longrightarrow X=X^T = 0$