# det

Determinant of symbolic matrix

## Description

example

B = det(A) returns the determinant of the square matrix of symbolic scalar variables A.

example

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

example

B = det(M) returns the determinant of the square symbolic matrix variable M. (since R2021a)

## Examples

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Compute the determinant of a matrix that contains symbolic scalar variables.

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

Compute the determinant of a matrix that contains 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 = $3 a {x}^{3}+6 {x}^{2}+4 x+2$

Since R2021a

This example shows how to compute the determinant of a block matrix. For example, find the determinant of a 4-by-4 block matrix

$\mathit{M}=\left[\begin{array}{cc}\mathbit{A}& 0\\ \mathbit{C}& \mathbit{B}\end{array}\right]$

where $A$, $B$, and $C$ are 2-by-2 submatrices.

Use symbolic matrix variables to represent the submatrices in the block matrix.

syms A B C [2 2] matrix
Z = symmatrix(zeros(2))
Z = ${\mathrm{0}}_{2,2}$
M = [A Z; C B]
M =

$\left(\begin{array}{c}\begin{array}{cc}A& {\mathrm{0}}_{2,2}\end{array}\\ \begin{array}{cc}C& B\end{array}\end{array}\right)$

Find the determinant of the matrix $M$.

det(M)
ans =

$\mathrm{det}\left(\begin{array}{c}\begin{array}{cc}A& {\mathrm{0}}_{2,2}\end{array}\\ \begin{array}{cc}C& B\end{array}\end{array}\right)$

Convert the result from symbolic matrix variable to symbolic scalar variables using symmatrix2sym.

D1 = simplify(symmatrix2sym(det(M)))
D1 = $\left({A}_{1,1} {A}_{2,2}-{A}_{1,2} {A}_{2,1}\right) \left({B}_{1,1} {B}_{2,2}-{B}_{1,2} {B}_{2,1}\right)$

Check if the determinant of matrix $M$ is equal to the determinant of $A$ times the determinant of $B$.

D2 = symmatrix2sym(det(A)*det(B))
D2 = $\left({A}_{1,1} {A}_{2,2}-{A}_{1,2} {A}_{2,1}\right) \left({B}_{1,1} {B}_{2,2}-{B}_{1,2} {B}_{2,1}\right)$
isequal(D1,D2)
ans = logical
1

## Input Arguments

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Input, specified as a square numeric matrix, or matrix of symbolic scalar variables.

Data Types: single | double | sym

Input, specified as a square symbolic matrix variable (since R2021a).

Data Types: symmatrix

## Tips

• Matrix computations involving many symbolic scalar variables can be slow. To increase the computational speed, reduce the number of symbolic scalar 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 scalar variables. This method is often suited to matrices that contain polynomial entries with multivariate coefficients.

## References

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