det

Determinant of symbolic matrix

Syntax

B = det(A)

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

Description

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

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

Examples

Compute Determinant of Symbolic Matrix

Compute the determinant of a symbolic matrix.

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

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

Input Arguments

A — Input matrix

square numeric matrix | square symbolic matrix

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.

References

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