Determinant

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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]

For a matrix , the determinant is the area of a parallellogram. (The area is equal to .)

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]

The determinant formula is a sum of products. Those products go along diagonals that "wrap around" to the top of the matrix. This trick is called the Rule of Sarrus.
  • 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]

  1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-09.
  2. 2.0 2.1 2.2 2.3 2.4 Weisstein, Eric W. "Determinant". mathworld.wolfram.com. Retrieved 2020-09-09.
  3. "Determinant of a Matrix". www.mathsisfun.com. Retrieved 2020-09-09.