# Matrix Determinants Calculator

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### Basic Information

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) $$det(AB) = det(A)det(B)$$.