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 A is written as \det(A) or |A| in a formula. Sometimes, instead of \det\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right) and \left|\begin{bmatrix}a&b\\c&d\end{bmatrix}\right|, one simply writes \det\begin{bmatrix}a&b\\c&d\end{bmatrix} and \left|\begin{matrix}a&b\\c&d\end{matrix}\right|.

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

### Geometric interpretation
An n \times n matrix can be seen as describing a linear map in n dimensions. In which case, the determinant indicates the factor by which this matrix scales (grows or shrinks) a region of n-dimensional space.

For example, a 2 \times 2 matrix A, seen as a linear map, will turn a square in 2-dimensional space into a parallelogram. That parallellogram's area will be \det(A) times as big as the square's area.

In the same way, a 3 \times 3 matrix B, seen as a linear map, will turn a cube in 3-dimensional space into a parallelepiped. That parallelepiped's volume will be \det(B) 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
:{\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = {\color{blue}{aei}+{dhc}+{gbf}}{\color{OrangeRed}{}-{gec}-{ahf}-{dbi}}}
:One can use the Rule of Sarrus (see image) to remember this formula.

### 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 n \times n matrix A. First, we choose any row or column of the matrix. For each number a_{ij} in that row or column, we calculate something called its **cofactor** C_{ij}. Then \det(A) = \sum a_{ij} C_{ij}.

To compute such a cofactor C_{ij}, we erase row i and column j from the matrix A. This gives us a smaller (n-1)\times(n-1) matrix. We call it M. The cofactor C_{ij} then equals (-1)^{i+j} \det(M).

Here is an example of a cofactor expansion of the left column of a 3 \times 3 matrix:

\begin{align}
\det \begin{bmatrix}{\color{red}1}&3&2\\{\color{red}2}&1&1\\{\color{red}0}&3&4\end{bmatrix}
&= {\color{red}1} \cdot C_{11} + {\color{red}2} \cdot C_{21} + {\color{red}0} \cdot C_{31}
\\ &= \left( {\color{red}1} \cdot (-1)^{1+1} \det\begin{bmatrix}1&1\\3&4\end{bmatrix} \right)
+ \left( {\color{red}2} \cdot (-1)^{2+1} \det\begin{bmatrix}3&2\\3&4\end{bmatrix} \right)
+ \left( {\color{red}0} \cdot (-1)^{3+1} \det\begin{bmatrix}3&2\\1&1\end{bmatrix} \right)
\\ &= ({\color{red}1} \cdot 1 \cdot 1) + ({\color{red}2} \cdot (-1) \cdot 6) + {\color{red}0}
\\ &= -11.
\end{align}

As illustrated above, one can simplify the computation of determinant by choosing a row or column that has many zeros; if a_{ij} is 0, then one can skip calculating C_{ij} altogether.

## Related pages
*Invertible matrix
*Volume

## References