Matrix Determinants Calculator

Pick the Size of the Matrices

2x2
3x3
4x4
5x5
6x6
7x7
8x8

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) \(det(AB) = det(A)det(B)\).
Our favorite Linear Algebra textbooks:
Book Cover
Schaum's Series - Linear Algebra
Buy from Amazon!
Book Cover
Steven Leon - Linear Algebra with Applications
Buy from Amazon!
Book Cover
D. C. Lay - Linear Algebra and its Applications
Buy from Amazon!
Book Cover
Gilbert Strang - Linear Algebra and Learning from Data
Buy from Amazon!