inv

Inverse of symbolic matrix

Syntax

D = inv(A)

Description

D = inv(A) returns the inverse of a symbolic matrix A.

Examples

Compute Inverse of Symbolic Matrix

Compute the inverse of a matrix of symbolic numbers.

A = sym([2 -1 0; -1 2 -1; 0 -1 2]);
D = inv(A)

D = 
$$\left(\begin{array}{ccc}\frac{3}{4}& \frac{1}{2}& \frac{1}{4}\\ \frac{1}{2}& 1& \frac{1}{2}\\ \frac{1}{4}& \frac{1}{2}& \frac{3}{4}\end{array}\right)$$

Compute the inverse of a matrix of symbolic scalar variables.

syms a b c d
A = [a b; c d];
D = inv(A)

D = 
$$\left(\begin{array}{cc}\frac{d}{a\hspace{0.17em}d-b\hspace{0.17em}c}& -\frac{b}{a\hspace{0.17em}d-b\hspace{0.17em}c}\\ -\frac{c}{a\hspace{0.17em}d-b\hspace{0.17em}c}& \frac{a}{a\hspace{0.17em}d-b\hspace{0.17em}c}\end{array}\right)$$

Compute Inverse of Hilbert Matrix with Symbolic Numbers

Compute the inverse of the Hilbert matrix that contains symbolic numbers.

D = inv(sym(hilb(4)))

D = 
$$\left(\begin{array}{cccc}16& -120& 240& -140\\ -120& 1200& -2700& 1680\\ 240& -2700& 6480& -4200\\ -140& 1680& -4200& 2800\end{array}\right)$$

Compute Inverse of Block Matrix

Find the inverse of a 4-by-4 block matrix

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

where $\mathit{A}$ and $\mathit{B}$ 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 [2 2] matrix
Z = symmatrix(zeros(2))

Z = $${\mathrm{0}}_{2,2}$$

C = [A Z; Z B]

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

Find the inverse of the matrix $\mathit{C}$.

D = inv(C)

D = 
$${\left(\begin{array}{c}\begin{array}{cc}A& {\mathrm{0}}_{2,2}\end{array}\\ \begin{array}{cc}{\mathrm{0}}_{2,2}& B\end{array}\end{array}\right)}^{-1}$$

To show the elements of the inverse matrix, convert the result from a symbolic matrix variable to symbolic scalar variables using symmatrix2sym.

D1 = symmatrix2sym(D)

D1 = 
$$\begin{array}{l}\left(\begin{array}{cccc}\frac{A_{2,2}}{{\sigma }_2}& -\frac{A_{1,2}}{{\sigma }_2}& 0& 0\\ -\frac{A_{2,1}}{{\sigma }_2}& \frac{A_{1,1}}{{\sigma }_2}& 0& 0\\ 0& 0& \frac{B_{2,2}}{{\sigma }_1}& -\frac{B_{1,2}}{{\sigma }_1}\\ 0& 0& -\frac{B_{2,1}}{{\sigma }_1}& \frac{B_{1,1}}{{\sigma }_1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=B_{1,1}\hspace{0.17em}{B}_{2,2}-B_{1,2}\hspace{0.17em}{B}_{2,1}\\ \\ \mathrm{ }{\sigma }_2=A_{1,1}\hspace{0.17em}{A}_{2,2}-A_{1,2}\hspace{0.17em}{A}_{2,1}\end{array}$$

Compute Inverse of Matrix Polynomial

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

fInv = inv(f)

fInv(a0, A) = $${\left(a_0\hspace{0.17em}{\mathrm{I}}_2+A\right)}^{-1}$$

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) = 
$$\begin{array}{l}\left(\begin{array}{cc}\frac{A_{2,2}+a_0}{{\sigma }_1}& -\frac{A_{1,2}}{{\sigma }_1}\\ -\frac{A_{2,1}}{{\sigma }_1}& \frac{A_{1,1}+a_0}{{\sigma }_1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=A_{1,1}\hspace{0.17em}{a}_0+A_{2,2}\hspace{0.17em}{a}_0+{a_0}^2+A_{1,1}\hspace{0.17em}{A}_{2,2}-A_{1,2}\hspace{0.17em}{A}_{2,1}\end{array}$$

Input Arguments

A — Input matrix

square numeric matrix | square matrix of symbolic scalar variables | square symbolic matrix variable | square symbolic matrix function | symbolic expression with square size

Input matrix, specified as a square numeric matrix, square matrix of symbolic scalar variables, square symbolic matrix variable, square symbolic matrix function, or symbolic expression with square size.

Data Types: single | double | sym | 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.

Version History

Introduced before R2006a

R2022a: Compute inverse of symbolic matrix functions

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

R2021b: Compute inverse of symbolic matrix variables

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