# Symbolic calculation of 3x3 determinants

## Online Calculator for Determinant 3x3

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

### Determinant

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

### 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.

### Example of the expansion according to the j-th row of a NxN determinant.

The Laplace expansion reduces the NxN determinant to a sum of (N-1)x(N-1) determinants.

$\mathrm{det A}=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{j1}& {a}_{j2}& \dots & {a}_{jn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|$

$=±{a}_{j1}|\begin{array}{ccc}{a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{\mathrm{j-1}2}& \dots & {a}_{\mathrm{j-1}n}\\ {a}_{\mathrm{j+1}2}& \dots & {a}_{\mathrm{j+1}n}\\ & ⋮\\ {a}_{n2}& \dots & {a}_{nn}\end{array}|±{a}_{j2}|\begin{array}{cccc}{a}_{11}& {a}_{13}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{\mathrm{j-1}1}& {a}_{\mathrm{j-1}3}& \dots & {a}_{\mathrm{j-1}n}\\ {a}_{\mathrm{j+1}1}& {a}_{\mathrm{j+1}3}& \dots & {a}_{\mathrm{j+1}n}\\ & ⋮\\ {a}_{n1}& {a}_{n3}& \dots & {a}_{nn}\end{array}|±\dots ±{a}_{jn}|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1\mathrm{n-1}}\\ & ⋮\\ {a}_{\mathrm{j-1}1}& {a}_{\mathrm{j-1}2}& \dots & {a}_{\mathrm{j-1}\mathrm{n-1}}\\ {a}_{\mathrm{j+1}1}& {a}_{\mathrm{j+1}2}& \dots & {a}_{\mathrm{j+1}\mathrm{n-1}}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{n\mathrm{n-1}}\end{array}|$

## More Calculators

Here is a list of of further useful calculators:

Index Matrix Determinant Determinant 2x2 Determinant 3x3 Determinant 3x3 symbolic Determinant 4x4 Determinant 4x4 symbolic Determinant 5x5 Determinant 5x5 symbolic Determinant NxN