<s>[INST] Find the determinants of the given matrices Consider scalars $a,b,c,d,e,f$ such that
$\det\left( \begin{array}{ccc}
a & 1 & d\\
b & 1 & e\\
c & 1 & f\\ \end{array} \right) = 7$ and $\det\left( \begin{array}{ccc}
a & 1 & d\\
b & 2 & e\\
c & 3 & f\\ \end{array} \right) = 11$
What is $\det\left( \begin{array}{ccc}
a & 3 & d\\
b & 3 & e\\
c & 3 & f\\ \end{array} \right)$ and $\det\left( \begin{array}{ccc}
a & 3 & d\\
b & 4 & e\\
c & 5 & f\\ \end{array} \right)$?
For the first one, I was thinking that the determinant would be $3\cdot 7 = 21$, and I think the determinant of the second one would still just be $11$ (since it's just a multiple of the second determinant matrix). Does this seem right? How can I solve these kinds of problems?
 [/INST] hint: $\left(\begin{array}{l} 3 \cr 4 \cr 5\end{array} \right) = 2 \left( \begin{array}{l} 1 \cr 1 \cr 1 \end{array} \right) + \left( \begin{array}{l} 1 \cr 2 \cr 3\end{array}\right).$
determinant is a linear function of every column. 
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