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# Determinant of a matrix

This online calculator computes determinant of a matrix, using it's definition

The site engine now allows input of long texts, so this is first calculator to use this feature, and it is used to enter matrix, and calculator computes the determinant of entered matrix.

It uses determinant definition, which is recursive calculation, and, in theory, quite resouce consuming, but for our case, with matrix, entered manually, I believe it is enough.

Some notes about determinant are below the calculator, for those who forgot.

### Determinant of a matrix

Digits after the decimal point: 2
Determinant of a matrix

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Some recursive formulae:

$det A=\begin{vmatrix} a_{11}\end{vmatrix} = a_{11}$

• determinant of a matrix 1x1

$det A=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}$

• determinant of a matrix 2x2

$det A=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$

• determinant of a matrix nxn, where n > 2
$\bar M_j^1$ - 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 recursive definition.

For example, here is the determinant of a matrix 3x3
$det A = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}-a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}+a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}$