<s>[INST] Given two determinants, find the det of this matrix? If det $ =\begin{bmatrix} a & 1 & d\\b & 1 & e\\c & 1 & f\\\end{bmatrix} = -2$
and If det $ =\begin{bmatrix} a & 1 & d\\b & 2 & e\\c & 3 & f\\\end{bmatrix} = -1$
what is the determinant of $\begin{bmatrix} a & -5 & d\\b & -7 & e\\c & -9 & f\\\end{bmatrix}$
I'm pretty lost here, I've tried multiplying matrices and subtracting them from each other but not getting the correct answer.
 [/INST] (This is essentially a restatement of the earlier answers, but expanding on how the multilinearity can be viewed.)
If we compute these determinants using an expansion along the second column, we find
$$\begin{vmatrix} a & 1 & d\\b & 1 & e\\c & 1 & f\\\end{vmatrix} = -1 \begin{vmatrix} b & e\\c & f\\\end{vmatrix}+1\begin{vmatrix} a & d\\c & f\\\end{vmatrix}-1\begin{vmatrix} a & d\\b & e\\\end{vmatrix}  =-2 \tag{eq 1}$$
and
$$\begin{vmatrix} a & 1 & d\\b & 2 & e\\c & 3 & f\\\end{vmatrix} = -1 \begin{vmatrix} b & e\\c & f\\\end{vmatrix}+2\begin{vmatrix} a & d\\c & f\\\end{vmatrix}-3\begin{vmatrix} a & d\\b & e\\\end{vmatrix}  =-1 \tag{eq 2}.$$
The determinant we want is
$$\begin{vmatrix} a & -5 & d\\b & -7 & e\\c & -9 & f\\\end{vmatrix} = -(-5) \begin{vmatrix} b & e\\c & f\\\end{vmatrix}+(-7)\begin{vmatrix} a & d\\c & f\\\end{vmatrix}-(-9)\begin{vmatrix} a & d\\b & e\\\end{vmatrix}$$
which is also given by $-3 \times (\text{eq } 1)-2 \times (\text{eq } 2).$  Hence the determinant is $8$.
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