Determinant

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The determinant of a square matrix is a scalar (a number) that tells you something about how that matrix behaves. You can calculate the determinant from the numbers in the matrix.

"The determinant of the matrix " is written as or in a formula. Sometimes, instead of and , we just write and .

Interpretation[change | change source]

There are a few ways to understand what the determinant says about the 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 this case, the determinant tells you 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. 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 the volume was mirrored over an odd number of axes.

"System of equations" interpretation[change | change source]

You can see a matrix as describing a system of linear equations. That system has a unique non-trivial solution exactly when the determinant is not 0. (Non-trivial means 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.

Geometrically, you 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, and 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, you can remember the formulas:

  • For matrices, the formula is:

    You 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.

Let's say 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 you can see here, we can save work by choosing a row or column that has many zeros. If is 0, we don't need to calculate .

See also[change | change source]