Determinant of a matrix

This paragraph describes a formula to compute the determinant of a 4-by-4 matrix using minors and cofactors of a matrix.

To compute the determinant of matrix $\mathbf{A}$ first one row or column is choosen. The sum of the four corresponding values of the row or column multiplied by the related cofactors results in the determinant:

$\det(\mathbf{A})=\sum_{(i,j) \isin \text{ one row or column}}{a_{i,j}C_{i,j}}$

 For the example matrix it is useful to choose the row 3 because it contains three zero values as factors:

 $\begin{align}
\det(\mathbf{A}_e)&=
\left|\begin{array}{cccc}
1 & 2 & 0 & 0\\
3 & 0 & 1 & 1\\
\mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0}\\
0 & 0 & 2 & 1
\end{array}\right| & \quad & \quad & \quad\\
&=
\mathbf{0}\cdot C_{3,1}&+&\mathbf{1}\cdot C_{3,2}&+&\mathbf{0}\cdot C_{3,3}&+&\mathbf{0}\cdot C_{3,4}\\
&=
0\cdot(-1)^{3+1}
\left|\begin{array}{ccc}
2 & 0 & 0\\
0 & 1 & 1\\
0 & 2 & 1
\end{array}\right|
&+&1\cdot(-1)^{3+2}
\left|\begin{array}{ccc}
1 & 0 & 0\\
3 & 1 & 1\\
0 & 2 & 1
\end{array}\right|
&+&0\cdot(-1)^{3+3}
\left|\begin{array}{ccc}
1 & 2 & 0\\
3 & 0 & 1\\
0 & 0 & 1
\end{array}\right|
&+&0\cdot(-1)^{3+4}
\left|\begin{array}{ccc}
1 & 2 & 0\\
3 & 0 & 1\\
0 & 0 & 2
\end{array}\right|\\
&= 0&+&1\cdot(-1)\cdot(-1)&+&0&+&0\\
&= 1&\quad&\quad&\quad
\end{align}$