## Determinant

Each n by n square matrix has 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 or.

When matrix order is 1, then

When the matrix order is 2,

, then

When the matrix order is 3,

, then

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

- The
**minor**of is the determinant of a sub matrix denoted by . The sub matrix obtained from matrix by deleting the row and column containing.

**Example**: we have then the sub matrix of is or and the minor is - The signed minor is called
**cofactor**of denoted by. Cofactor is a scalar sign determined by . This sign follows the following order

**Example: **

Suppose we have then the cofactor of is

The **determinant of matrix** is the sum product of the first column entries with its cofactor, that is

**For example**,

, then

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.

## Properties

The following are well known properties of matrix determinant:

- A square matrix is singular (has no inverse) if and only if
- If a square matrix has an inverse, the determinant of an inverse matrix is the reciprocal of the matrix determinant, that is.
- When matrix has a row or a column consisting entirely of zeros, then
- When matrix has two identical rows or two identical columns, then
- Determinant of a matrix product is equal to the product of their determinant,
- Determinant does not change when we transpose the matrix,
- 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 matrixis an triangular matrix, the determinant equals the product of the diagonal elements of

**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. https:\\people.revoledu.com\kardi\tutorial\LinearAlgebra\