# Symbolic calculation of 3x3 determinants

## Calculator

The online calculators are to calculate the determinat of 2x2, 3x3, 4x4,... and nxn matrices.

Online Calculators:

## Online Calculator for Determinant 3x3

The online calculator calculates symbolic the value of the determinant of a 3x3 matrix after Sarrus rule and with the Laplace expansion in a row or column.

### Determinant

$\mathrm{det A}=\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|$

### Enter the coefficients

Brackets has to be set explicit. Not a+b but (a+b) is ok.

a11= a12= a13=

a21= a22= a23=

a31= a32= a33=

### Calculation using the Laplace expansion

You can select the row or column to be used for expansion.

### Laplace Expansion Theorem

The Laplacian development theorem provides a method for calculating the determinant, in which the determinant is developed after a row or column. The dimension is reduced and can be reduced further step by step up to a scalar.

$\mathrm{det A}=\sum _{i=1}^{n}{-1}^{i+j}\cdot {a}_{ij}\mathrm{det}{A}_{ij}\text{( Expansion on the j-th column )}$

$\mathrm{det A}=\sum _{j=1}^{n}{-1}^{i+j}\cdot {a}_{ij}\mathrm{det}{A}_{ij}\text{( Expansion on the i-th row )}$

where Aij, the sub-matrix of A, which arises when the i-th row and the j-th column are removed.