Which determinant could we find?

Question

$A$ and $B$ are matrices and I found the determinants of

$$A + B,\, A - B,\, AB,\, A^{-1},\, B^T.$$

If we know the determinants of $A$ and $B$ but don't remember the matrices $A$ and $B$, which of the determinants above could we have found?

I think it's only $B^T$ and $A^{-1}$. Can someone tell me if that is correct?

Answer 1

From some basic facts about determinants we know that:

• $|B|=|B^T|$ (ProofWiki, Google)
• $|AB|=|A|\cdot|B|$ (ProofWiki, Google)
• $|A^{-1}|=|A|^{-1}$ (ProofWiki, Google)

So if we know $|A|$ and $|B|$, we also know the determinants of $AB$, $B^T$ and $A^{-1}$.

To show that we cannot say anything about the value of $|A\pm B|$ from the values of $|A|$ and $|B|$, we can simply try some examples:

• For $A=B=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ we have $|A|=|B|=1$ and $|A+B|=4$ and $|A-B|=0$.
• For $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and $B=\begin{pmatrix}1&0\\1&1\end{pmatrix}$ we again have $|A|=|B|=1$. But in this case $|A+B|=\begin{vmatrix}2&1\\1&2\end{vmatrix}=3$ and $|A-B|=\begin{vmatrix}0&1\\-1&0\end{vmatrix}=1$.

So in both cases $|A|=|B|=1$, but determinants of $A\pm B$ have different values. Which means that they are not determined by the values of $|A|$ and $|B|$.