Definition: The determinant of the square matrix \(A=[a_{ij}]\), denoted by \(det(A)\) or \(|A|\), is defined as:
\(\begin{vmatrix}
a_{11} & a_{12} & \dots & a_{1n}\\
a_{21} & a_{22} & \dots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{n1} & a_{n2} & \dots & a_{nn}
\end{vmatrix}=\)
\(a_{11} \begin{vmatrix}
a_{22} & a_{23} & \dots & a_{2n}\\
a_{32} & a_{33} & \dots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{n2} & a_{n3} & \dots & a_{nn}
\end{vmatrix}-\)
\(a_{12} \begin{vmatrix}
a_{21} & a_{23} & \dots & a_{2n}\\
a_{31} & a_{33} & \dots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{n1} & a_{n3} & \dots & a_{nn}
\end{vmatrix}+\)
\(\dots \pm \)
\(a_{1n} \begin{vmatrix}
a_{21} & a_{22} & \dots & a_{2(n-1)}\\
a_{31} & a_{32} & \dots & a_{3(n-1)}\\
\vdots & \vdots & \ddots & \vdots\\
a_{n1} & a_{n2} & \dots & a_{n(n-1)}
\end{vmatrix}\)
Here are a few theorems that may facilitate calculations of matrix determinants. Let \(A\) be a square matrix :
a) The determinant of \(A\) and the determinant of its transpose are the same \(det(A)=det(A^T)\);
b) If \(A\) has a row or column of zeros, then \( det(A) = 0 \);
c) If \(A\) has two identical rows (or columns), then \( det(A) = 0 \);
d) \(det(AB) = det(A)det(B)\).
