Determinant facts for kids
The determinant of the matrix is written as or in a formula. Sometimes, instead of and , one simply writes and .
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.
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.
Determinant Facts for Kids. Kiddle Encyclopedia.