Determinant

sum of signed terms of n factors from n×n matrix with no two factors sharing row or column
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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 .

InterpretationEdit

There are a few ways to understand what the determinant says about a matrix.

Geometric interpretationEdit

 
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" interpretationEdit

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 matricesEdit

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 determinantEdit

There are a few ways to calculate a determinant.

Formulas for small matricesEdit

 
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 expansionEdit

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 pagesEdit

ReferencesEdit

  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.