<s>[INST] What would the determinant of the following matrix be? " If $$\det\begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}=4$$
and $$\det
\begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}=3$$
What is $$\det \begin{pmatrix}a&-1&d\\ b&-3&e\\ c&-5&f\end{pmatrix}?$$"
This does not seem to fit into any of the regular changes in the values of matrices. A row does not seem to be multiplied. It does not seem like a row is being added to another. So how would I find the required determinant?
 [/INST] $\det \begin{pmatrix}a&-1&d\\ b&-3&e\\ c&-5&f\end{pmatrix}=\det\begin{pmatrix}a&1-2\times1&d\\ b&1-2\times 2&e\\ c&1-2\times 3&f\end{pmatrix}=\det \begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}-2\det \begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}$
</s>