# det

Determinant of symbolic matrix

## Syntax

``B = det(A)``
``B = det(A,'Algorithm','minor-expansion')``

## Description

example

````B = det(A)` returns the determinant of the square matrix `A`.```

example

````B = det(A,'Algorithm','minor-expansion')` uses the minor expansion algorithm to evaluate the determinant of `A`.```

## Examples

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Compute the determinant of a symbolic matrix.

```syms a b c d M = [a b; c d]; B = det(M)```
`B = $a d-b c$`

Compute the determinant of a matrix that contain symbolic numbers.

```A = sym([2/3 1/3; 1 1]); B = det(A)```
```B =  $\frac{1}{3}$```

Create a symbolic matrix that contains polynomial entries.

```syms a x A = [1, a*x^2+x, x; 0, a*x, 2; 3*x+2, a*x^2-1, 0]```
```A =  $\left(\begin{array}{ccc}1& a {x}^{2}+x& x\\ 0& a x& 2\\ 3 x+2& a {x}^{2}-1& 0\end{array}\right)$```

Compute the determinant of the matrix using minor expansion.

`B = det(A,'Algorithm','minor-expansion')`
`B = $3 a {x}^{3}+6 {x}^{2}+4 x+2$`

## Input Arguments

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Input, specified as a square numeric or symbolic matrix.

## Tips

• Matrix computations involving many symbolic variables can be slow. To increase the computational speed, reduce the number of symbolic variables by substituting the given values for some variables.

• The minor expansion method is generally useful to evaluate the determinant of a matrix that contains many symbolic variables. This method is often suited to matrices that contain polynomial entries with multivariate coefficients.

 Khovanova, T. and Z. Scully. "Efficient Calculation of Determinants of Symbolic Matrices with Many Variables." arXiv preprint arXiv:1304.4691 (2013).

## Support

#### Mathematical Modeling with Symbolic Math Toolbox

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