<s>[INST] Determinants of $3\times 3$ Matrix Suppose that $a,b,c,d,e,f$  are numbers such that 
$$\det\left(\begin{matrix}
a&1&d\\b&1&e\\ c&1&f
\end{matrix}\right)=7$$ and 
$$\det\left(\begin{matrix}
a&1&d\\b&2&e\\ c&3&f
\end{matrix}\right)=11.$$
How do you find the determinant of the Matrix 
$$\begin{pmatrix}
a&3&d\\b&5&e\\ c&7&f
\end{pmatrix}?$$
Any suggestions on how to approach the question would be greatly appreciated.
 [/INST] Using multilinearity of the determinant (by columns), we get:
$$\begin{vmatrix}
a&3&d\\b&5&e\\ c&7&f
\end{vmatrix}=\begin{vmatrix}
a&1+2&d\\b&1+4&e\\ c&1+6&f
\end{vmatrix}=\begin{vmatrix}
a&1&d\\b&1&e\\ c&1&f
\end{vmatrix}+2\begin{vmatrix}
a&1&d\\b&2&e\\ c&3&f
\end{vmatrix}=7+2\cdot 11=29$$
</s>