If A and B are n×n matrices, then |A+B^T |=|A^T+B|

Question

If $A$ and $B$ are $n×n$ matrices, then $|A+B^T|=|A^T+B|$

can I prove that it's t/f without giving a value to the matrices?

Answer 1

Nice two-liner solves this for you: $$\mathrm{det}(A) = \mathrm{det}(A^T)$$ $$\mathrm{det}(A+B^T) = \mathrm{det}([A+B^T])^T) = \mathrm{det}(A^T+B)$$

You can show $(A+B)^T = A^T + B^T$ in the middle there for a more complete proof.