# What is the determinant of a matrix?

Mar 10, 2016

See explanation...

#### Explanation:

An $n \times n$ matrix represents a linear transformation (reflection, rotation, stretching, shearing, etc.) in $n$ dimensional space.

The determinant is basically the factor by which a matrix scales the area/volume/hypervolume - with negative value if reflection is involved. If the result is zero then the matrix collapses $n$ dimensions to fewer dimensions and the matrix is not invertible.

For example, in $2$ dimensions, the determinant of a matrix $M$ is the area of the quadrilateral obtained by transforming the unit square by multiplying its coordinates by $M$.

For $2 \times 2$ matrices we can calculate the determinant as follows:

$\left\mid \begin{matrix}a & b \\ c & d\end{matrix} \right\mid = a d - b c$

For $3 \times 3$ matrices we can calculate the determinant as follows:

$\left\mid \begin{matrix}{a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33}\end{matrix} \right\mid = {a}_{11} \left\mid \begin{matrix}{a}_{22} & {a}_{23} \\ {a}_{32} & {a}_{33}\end{matrix} \right\mid + {a}_{12} \left\mid \begin{matrix}{a}_{23} & {a}_{21} \\ {a}_{33} & {a}_{31}\end{matrix} \right\mid + {a}_{13} \left\mid \begin{matrix}{a}_{21} & {a}_{22} \\ {a}_{31} & {a}_{32}\end{matrix} \right\mid$

A similar process can be used on larger matrices, but there are quicker methods.