How do you solve a determinant thats set to a value with 1 unknown variable?

Question

I was just wondering if someone could explain the steps you take to solve a determinant that has an unknown variable, and is set to equal integer value? For example:

How is one supposed to go about isolating the variable 'a' so that you can obtain its values?

Thanks heaps in advance Tim

Answer 1

Hint: $\det\begin{pmatrix}7&1&3&-2\\-2&1&-12&-1\\1&6&-4&a\\2&4&2&1\end{pmatrix}=\det\begin{pmatrix}7&1&3&-2\\-2&1&-12&-1\\1&6&-4&0\\2&4&2&1\end{pmatrix}+\det\begin{pmatrix}7&1&3&0\\-2&1&-12&0\\1&6&-4&a\\2&4&2&0\end{pmatrix}=\det\begin{pmatrix}7&1&3&-2\\-2&1&-12&-1\\1&6&-4&0\\2&4&2&1\end{pmatrix}-a\cdot\det\begin{pmatrix}7&1&3\\-2&1&-12\\2&4&2\end{pmatrix}$.

Answer 2

You can expand along the fourth column to get an equation which is like $2* det(3x3) - det(3x3) -a det(3x3) +det(3x3) = 2015$ where the 3x3 matrices are formed by deleting the appropriate row and column. Then, calculate all the determinants and solve for $a$.

Answer 3

Evaluating the determinat you find $1415-300 \alpha=2015\;$, so: $\alpha=-2$.