# Matrix Determinant Calculator

The calculator computes the determinant of a matrix.

### This page exists due to the efforts of the following people:

#### Anton

Created: 2011-06-24 21:29:11, Last updated: 2023-04-02 15:29:12

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You should enter the square matrix, and the calculator computes the determinant of the entered matrix.

It uses the definition of a determinant, which is a recursive calculation, and, in theory, quite resource consuming. Still, for our case, with matrixes entered manually, I believe it is enough (if not, please use Determinant by Gaussian elimination).

#### Determinant of a matrix

Digits after the decimal point: 2
Determinant of a matrix

### Determinant of a matrix

• determinant of a matrix 1x1
$det A=\begin{vmatrix} a_{11}\end{vmatrix} = a_{11}$

• determinant of a matrix 2x2
$det A=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}$

• determinant of a matrix nxn, where n > 2
$det A=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$

where

$\bar M_j^1$ - minor of $a_{1j}$.

Minor of $a_{1j}$ - is the determinant of a (n–1) × (n–1) matrix that results from deleting the 1-th row and the j-th column of A.

That's why this is a recursive definition. For example, here is the determinant of a matrix 3x3

$det A = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}-a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}+a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}$

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