Matrix Determinants Calculator

Definition: The determinant of the square matrix $A=[a_{ij}]$, denoted by $det(A)$ or $|A|$, is defined as:

$\left|\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \dots & a_{nn}\end{array}\right|=$ $a_{11}\left|\begin{array}{cccc}{a}_{22}& a_{23}& \dots & a_{2n}\\ a_{32}& a_{33}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n2}& a_{n3}& \dots & a_{nn}\end{array}\right|-$
$a_{12}\left|\begin{array}{cccc}{a}_{21}& a_{23}& \dots & a_{2n}\\ a_{31}& a_{33}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n3}& \dots & a_{nn}\end{array}\right|+$ $\cdots \pm$ $a_{1n}\left|\begin{array}{cccc}{a}_{21}& a_{22}& \dots & a_{2(n-1)}\\ a_{31}& a_{32}& \dots & a_{3(n-1)}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \dots & a_{n(n-1)}\end{array}\right|$ 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) $det(AB)=det(A)det(B)$.