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

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

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.

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

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

where A_ij, 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.

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

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

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