det

Determinant of symbolic matrix

Syntax

B = det(A)

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

B = det(M)

Description

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

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

B = det(M) returns the determinant of the square symbolic matrix variable or matrix function M.

Examples

Compute Determinant of Matrix with Symbolic Scalar Variables

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 = $$a\hspace{0.17em}d-b\hspace{0.17em}c$$

Compute Determinant of Matrix with Symbolic Numbers

Compute the determinant of a matrix that contains symbolic numbers.

A = sym([2/3 1/3; 1 1]);
B = det(A)

B = 
$$\frac{1}{3}$$

Compute Determinant Using Minor Expansion

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 = 
$$\left(\begin{array}{ccc}1& a\hspace{0.17em}{x}^2+x& x\\ 0& a\hspace{0.17em}x& 2\\ 3\hspace{0.17em}x+2& a\hspace{0.17em}{x}^2-1& 0\end{array}\right)$$

Compute the determinant of the matrix using minor expansion.

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

B = $$3\hspace{0.17em}a\hspace{0.17em}{x}^3+6\hspace{0.17em}{x}^2+4\hspace{0.17em}x+2$$

Compute Determinant of Block Matrix

Compute the determinant of a 4-by-4 block matrix

$$\mathit{M}=\left[\begin{array}{cc}\mathit{A}& 0_{2,2}\\ \mathit{C}& \mathit{B}\end{array}\right]$$

where $$A$$, $$B$$, and $$C$$ are 2-by-2 submatrices. The notation $0_{2,2}$ represents a 2-by-2 submatrix of zeros.

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

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

D1 = simplify(symmatrix2sym(det(M)))

D1 = $$\left(A_{1,1}\hspace{0.17em}{A}_{2,2}-A_{1,2}\hspace{0.17em}{A}_{2,1}\right)\hspace{0.17em}\left(B_{1,1}\hspace{0.17em}{B}_{2,2}-B_{1,2}\hspace{0.17em}{B}_{2,1}\right)$$

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 = $$\left(A_{1,1}\hspace{0.17em}{A}_{2,2}-A_{1,2}\hspace{0.17em}{A}_{2,1}\right)\hspace{0.17em}\left(B_{1,1}\hspace{0.17em}{B}_{2,2}-B_{1,2}\hspace{0.17em}{B}_{2,1}\right)$$

isequal(D1,D2)

ans = logical
   1

Compute Determinant of Matrix Polynomial

Compute the determinant of the matrix polynomial ${\mathit{a}}_0{\mathit{I}}_2+\mathit{A}$, where $\mathit{A}$ is a 2-by-2 matrix.

Create the matrix $\mathit{A}$ as a symbolic matrix variable and the coefficient ${\mathit{a}}_0$ as a symbolic scalar variable. Create the matrix polynomial as a symbolic matrix function f with ${\mathit{a}}_0$ and $\mathit{A}$ as its parameters.

syms A [2 2] matrix
syms a0
syms f(a0,A) [2 2] matrix keepargs
f(a0,A) = a0*eye(2) + A

f(a0, A) = $$a_0\hspace{0.17em}{\mathrm{I}}_2+A$$

Find the determinant of f using det. The result is a symbolic matrix function of type symfunmatrix that accepts scalars, vectors, and matrices as its input arguments.

fInv = det(f)

fInv(a0, A) = $$\det \left(a_0\hspace{0.17em}{\mathrm{I}}_2+A\right)$$

Convert the result from the symfunmatrix data type to the symfun data type using symfunmatrix2symfun. The result is a symbolic function that accepts scalars as its input arguments.

gInv = symfunmatrix2symfun(fInv)

gInv(a0, A1_1, A1_2, A2_1, A2_2) = $$A_{1,1}\hspace{0.17em}{a}_0+A_{2,2}\hspace{0.17em}{a}_0+{a_0}^2+A_{1,1}\hspace{0.17em}{A}_{2,2}-A_{1,2}\hspace{0.17em}{A}_{2,1}$$

Input Arguments

A — Input matrix

square matrix of symbolic numbers | square matrix of symbolic scalar variables | square matrix of symbolic functions

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

Data Types: single | double | sym | symfun

M — Input matrix

square symbolic matrix variable | square symbolic matrix function

Input matrix, specified as a square symbolic matrix variable or square symbolic matrix function.

Data Types: symmatrix | symfunmatrix

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 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).

Version History

Introduced before R2006a

R2022a: Compute determinant of symbolic matrix functions

The det function accepts an input argument of type symfunmatrix. For an example, see Compute Determinant of Matrix Polynomial.

R2021a: Compute determinant of symbolic matrix variables

The det function accepts an input argument of type symmatrix. For an example, see Compute Determinant of Block Matrix.