inv

Inverse of symbolic matrix

Syntax

D = inv(A)

D = inv(M)

Description

D = inv(A) returns the inverse of the square matrix of symbolic scalar variables A.

D = inv(M) returns the inverse of the square symbolic matrix variable M. (since R2021b)

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

Since R2021b

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

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

where $\mathit{A}$ and $\mathit{B}$ are 2-by-2 submatrices. The notation $$0$$ 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 = 
$$\begin{array}{l}\left(\begin{array}{cccc}\frac{A_{2,2}}{{\sigma }_2}& -\frac{A_{1,2}}{{\sigma }_2}& 0& 0\\ -\frac{A_{2,1}}{{\sigma }_2}& \frac{A_{1,1}}{{\sigma }_2}& 0& 0\\ 0& 0& \frac{B_{2,2}}{{\sigma }_1}& -\frac{B_{1,2}}{{\sigma }_1}\\ 0& 0& -\frac{B_{2,1}}{{\sigma }_1}& \frac{B_{1,1}}{{\sigma }_1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=B_{1,1}\hspace{0.17em}{B}_{2,2}-B_{1,2}\hspace{0.17em}{B}_{2,1}\\ \\ \mathrm{ }{\sigma }_2=A_{1,1}\hspace{0.17em}{A}_{2,2}-A_{1,2}\hspace{0.17em}{A}_{2,1}\end{array}$$

Input Arguments

A — Input matrix

square numeric matrix | square matrix of symbolic scalar variables

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

Data Types: single | double | sym

M — Input matrix

square symbolic matrix variable

Since R2021b

Input matrix, specified as a square symbolic matrix variable.

Data Types: symmatrix

Limitations

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.