By Kardi Teknomo, PhD.

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Each n by n square matrix Determinanthas an associated scalar number, we called as determinant of that matrix, whose value will determine whether the matrix has an inverse or not. The symbol of determinant is either DeterminantorDeterminant.

When matrix order is 1, Determinant then Determinant

When the matrix order is 2,
Determinant, then Determinant

When the matrix order is 3,
Determinant, then Determinant


When we have a matrixDeterminant then we can define several things:

  1. The minor of Determinant is the determinant of a Determinant sub matrix denoted by Determinant. The sub matrix Determinantobtained from matrixDeterminant by deleting the row and column containingDeterminant.
    Example: we have Determinantthen the sub matrix of Determinant is Determinant orDeterminant and the minor is Determinant
  2. The signed minor is called cofactor of Determinant denoted byDeterminant. Cofactor is a scalar sign determined byDeterminant . This sign follows the following order


Suppose we have Determinantthen the cofactor of Determinant is

The determinant of matrix Determinant is the sum product of the first column entries with its cofactor, that is Determinant
For example,
Determinant, then Determinant Determinant

Hand (manual) computation of determinant of matrix size larger than 3 is quite time consuming. The interactive program below will help you to compute determinant of your square matrix of any positive order less than 10. Random Example button will generate random square matrix of random size.


The following are well known properties of matrix determinant:

  • A square matrix Determinantis singular (has no inverse) if and only if Determinant
  • If a square matrix Determinanthas an inverse, the determinant of an inverse matrix is the reciprocal of the matrix determinant, that isDeterminant.
  • When matrix Determinanthas a row or a column consisting entirely of zeros, then Determinant
  • When matrix Determinanthas two identical rows or two identical columns, then Determinant
  • Determinant of a matrix product is equal to the product of their determinant, Determinant
  • Determinant does not change when we transpose the matrix, Determinant
  • Determinant does not change when we add a multiple of a row or a column to another.
  • Interchanging two rows or two columns of a matrix changes the sign of the determinant but the magnitude of the determinant does not change.
  • Multiplying a row or a column of a matrix by a scalar will make the value of the determinant multiplied by that scalar. Similarly, removing a common factor from a row or a column will also remove the common factor from its determinant.
  • When matrixDeterminantis anDeterminant triangular matrix, the determinant equals the product of the diagonal elements ofDeterminant

See also: matrix product, matrix inverse, matrix transpose, and elementary row operations, matrix trace, matrix rank, Rotation or Reflection

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Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\\kardi\tutorial\LinearAlgebra\