# inv

Inverse of symbolic matrix

## Syntax

``D = inv(A)``
``D = inv(M)``

## Description

example

````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

collapse all

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

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

`D = inv(sym(hilb(4)))`
```D =  $\left(\begin{array}{cccc}16& -120& 240& -140\\ -120& 1200& -2700& 1680\\ 240& -2700& 6480& -4200\\ -140& 1680& -4200& 2800\end{array}\right)$```

Since R2021b

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

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

where $\mathbit{A}$ and $\mathbit{B}$ 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 $\mathbit{C}$.

`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 =  ```

## Input Arguments

collapse all

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

Data Types: `single` | `double` | `sym`

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.