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.


det A= | a11a12a13 a21a22a23 a31a32a33 |

Enter the coefficients

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


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.

det A= i = 1 n -1 i + j a i j det A i j ( Expansion on the j-th column )

det A= j = 1 n -1 i + j a i j det A i j ( 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.

det A= | a11a12a1n aj1aj2ajn an1an2ann |

=±aj1 | a12a1n aj-12aj-1n aj+12aj+1n an2ann | ±aj2 | a11a13a1n aj-11aj-13aj-1n aj+11aj+13aj+1n an1an3ann | ±±ajn | a11a12a1n-1 aj-11aj-12aj-1n-1 aj+11aj+12aj+1n-1 an1an2ann-1 |

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