Let me start off by saying that I realise that I need to apply the following theorems:
If $B$ is a matrix that results when a single row or single column of $A$ is multiplied by a scalar $k$, then $\begin{vmatrix}B\end{vmatrix}=k\begin{vmatrix}A\end{vmatrix}$.
If $B$ is a matrix that results when two rows or two columns of $A$ are interchanged, then $\begin{vmatrix}B\end{vmatrix}=-\begin{vmatrix}A\end{vmatrix}$.
If $B$ is a matrix that results when a multiple of one row of $A$ is added to another row or when a multiple of one column is added to another column, then $\begin{vmatrix}B\end{vmatrix}=\begin{vmatrix}A\end{vmatrix}$.
I am just unable to translate them into application. I can see similarities between the matrices in my problem, but I am not quite sure how to actually solve it
The problem is as follows:
$$Given~~\begin{vmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \\ \end{vmatrix} = 11, \begin{vmatrix} a & 1 & d \\ b & 1 & e \\ c & 1 & f \\ \end{vmatrix} = 7\\ Find \begin{vmatrix} a & 3 & d \\ b & 2 & e \\ c & 3 & f \\ \end{vmatrix} $$
I have tried building a system of linear equations, namely:
$$c_1 * u + c_2 * v + c_3 * w = |M|$$ where $c_1$, $c_2$ and $c_3$ are the entries down the second column, and $u, v$ and $w$ are substitutes for the determinants of the matrix of minors, and $|M|$ is the determinant of the matrix. I did it regardless of the fact that I knew I would land up with 3 equations in 4 unknowns, because I thought I could do some magic with ratios of $u:v:w$, but that is not the case.
Any assistance would be appreciated.
The problem is under determined.
If you choose $c=3,d=2,e=1$, $a={10 f + 20 \over f^2 }$, $b={10 \over f^2}$, the determinants of the first two matrices are $11,7$ as expected, and the determinant of the last matrix is $17-{20 \over f}$.