Determinant
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.[1][2] Sometimes, instead of and , one simply writes and .
Interpretation
[change | change source]There are a few ways to understand what the determinant says about a matrix.
Geometric interpretation
[change | change source]![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/220px-Area_parallellogram_as_determinant.svg.png)
An matrix can be seen as describing a linear map in dimensions. In which case, the determinant indicates the factor by which this matrix scales (grows or shrinks) a region of -dimensional space.
For example, a matrix , seen as a linear map, will turn a square in 2-dimensional space into a parallelogram. That parallellogram's area will be times as big as the square's area.
In the same way, a matrix , seen as a linear map, will turn a cube in 3-dimensional space into a parallelepiped. That parallelepiped's volume will be times as big as the cube's volume.
The determinant can be negative or zero. A linear map can stretch and scale a volume, but it can also reflect it over an axis. Whenever this happens, the sign of the determinant changes from positive to negative, or from negative to positive. A negative determinant means that the volume was mirrored over an odd number of axes.
"System of equations" interpretation
[change | change source]One can think of a matrix as describing a system of linear equations. That system has a unique non-trivial solution exactly when the determinant is not 0[2] (non-trivial meaning that the solution is not just all zeros).
If the determinant is zero, then there is either no unique non-trivial solution, or there are infinitely many.
Singular matrices
[change | change source]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.[2]
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
[change | change source]There are a few ways to calculate a determinant.
Formulas for small matrices
[change | change source]![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Sarrus_rule_vertical.svg/220px-Sarrus_rule_vertical.svg.png)
- For and matrices, the following simple formulas hold:[2]
- For matrices, the formula is:[3]
- One can use the Rule of Sarrus (see image) to remember this formula.
Cofactor expansion
[change | change source]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 .[2]
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
[change | change source]References
[change | change source]- ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-09.
- ↑ 2.0 2.1 2.2 2.3 2.4 Weisstein, Eric W. "Determinant". mathworld.wolfram.com. Retrieved 2020-09-09.
- ↑ "Determinant of a Matrix". www.mathsisfun.com. Retrieved 2020-09-09.