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 formula_1 is written as formula_2 or formula_3 in a formula. Sometimes, instead of formula_4 and formula_5, one simply writes formula_6 and formula_7.
Interpretation.
There are a few ways to understand what the determinant says about a matrix.
Geometric interpretation.
An formula_8 matrix can be seen as describing a linear map in formula_9 dimensions. In which case, the determinant indicates the factor by which this matrix scales (grows or shrinks) a region of formula_9-dimensional space.For example, a formula_11 matrix formula_1, seen as a linear map, will turn a square in 2-dimensional space into a parallelogram. That parallellogram's area will be formula_2 times as big as the square's area.In the same way, a formula_14 matrix formula_15, seen as a linear map, will turn a cube in 3-dimensional space into a parallelepiped. That parallelepiped's volume will be formula_16 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.
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 ("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.
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.
Cofactor expansion.
For larger matrices, the determinant is harder to calculate. One way to do it is called cofactor expansion.
Suppose that we have an formula_8 matrix formula_1. First, we choose any row or column of the matrix. For each number formula_24 in that row or column, we calculate something called its cofactor formula_25. Then formula_26.
To compute such a cofactor formula_25, we erase row formula_28 and column formula_29 from the matrix formula_1. This gives us a smaller formula_31 matrix. We call it formula_32. The cofactor formula_25 then equals formula_34.
Here is an example of a cofactor expansion of the left column of a formula_14 matrix:
formula_36
As illustrated above, one can simplify the computation of determinant by choosing a row or column that has many zeros; if formula_24 is 0, then one can skip calculating formula_25 altogether.