# Determinant

sum of signed terms of n factors from n×n matrix with no two factors sharing row or column
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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.[1][2] 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

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 ${\displaystyle n}$  dimensions. In which case, the determinant indicates the factor by which this matrix scales (grows or shrinks) a region of ${\displaystyle n}$ -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[2] (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.[2]

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.

### 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:[2]

${\displaystyle \det {\begin{bmatrix}a\end{bmatrix}}=a,\qquad \det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.}$

• For ${\displaystyle 3\times 3}$  matrices, the formula is:[3]

${\displaystyle {\det {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}={\color {blue}{aei}+{dhc}+{gbf}}{\color {OrangeRed}{}-{gec}-{ahf}-{dbi}}}}$

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 ${\displaystyle \det(A)=\sum a_{ij}C_{ij}}$ .[2]

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.

## References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-09.
2. Weisstein, Eric W. "Determinant". mathworld.wolfram.com. Retrieved 2020-09-09.
3. "Determinant of a Matrix". www.mathsisfun.com. Retrieved 2020-09-09.