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

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

### Geometric interpretation

For a {\displaystyle 2\times 2} matrix {\displaystyle {\begin{bmatrix}a&c\\b&d\end{bmatrix}}}, the determinant is the area of a parallellogram. (The area is equal to {\displaystyle ad-bc}.)

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

In the same way, a {\displaystyle 3\times 3} matrix {\displaystyle B}, seen as a linear map, will turn a cube in 3-dimensional space into a parallelepiped. That parallelepiped's volume will be {\displaystyle \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 (*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
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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
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There are a few ways to calculate a determinant.

### Formulas for small matrices

The {\displaystyle 3\times 3} 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 {\displaystyle 1\times 1} and {\displaystyle 2\times 2} matrices, the following simple formulas hold:{\displaystyle \det {\begin{bmatrix}a\end{bmatrix}}=a,\qquad \det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.}

* For {\displaystyle 3\times 3} matrices, the formula is:{\displaystyle {\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 {\displaystyle n\times n} matrix {\displaystyle A}. First, we choose any row or column of the matrix. For each number {\displaystyle a\_{ij}} in that row or column, we calculate something called its **cofactor** {\displaystyle C\_{ij}}. Then {\displaystyle \det(A)=\sum a\_{ij}C\_{ij}}.

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

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

{\displaystyle {\begin{aligned}\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{aligned}}}

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

Related pages
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* Invertible matrix
* Volume