# Determinant facts for kids

The **determinant** of a square matrix is a scalar (a number) that indicates how that matrix behaves. It can be calculated from the numbers in the matrix.

The determinant of the matrix is written as or in a formula. Sometimes, instead of and , one simply writes and .

## Contents

## Singular matrices

A matrix has an inverse matrix exactly when the determinant is not 0. For this reason, a matrix with a non-zero determinant is called **invertible**. If the determinant is 0, then the matrix is called **non-invertible** or **singular**.

Geometrically, one can think of a singular matrix as "flattening" the parallelepiped into a parallelogram, or a parallelogram into a line. Then the volume or area is 0, which means that there is no linear map that will bring the old shape back.

## Calculating a determinant

There are a few ways to calculate a determinant.

### Formulas for small matrices

- For and matrices, the following simple formulas hold:

- For matrices, the formula is:
One can use the Rule of Sarrus (see image) to remember this formula.

### Cofactor expansion

For larger matrices, the determinant is harder to calculate. One way to do it is called **cofactor expansion**.

Suppose that we have an matrix . First, we choose any row or column of the matrix. For each number in that row or column, we calculate something called its **cofactor** . Then .

To compute such a cofactor , we erase row and column from the matrix . This gives us a smaller matrix. We call it . The cofactor then equals .

Here is an example of a cofactor expansion of the left column of a matrix:

As illustrated above, one can simplify the computation of determinant by choosing a row or column that has many zeros; if is 0, then one can skip calculating altogether.

## Related pages

## See also

In Spanish: Determinante (matemática) para niños

Delilah Pierce |

Gordon Parks |

Augusta Savage |

Charles Ethan Porter |

*Kiddle Encyclopedia.*