linsolve

Solve symbolic linear equations in matrix form

Syntax

X = linsolve(A,B)

[X,R] = linsolve(A,B)

Description

X = linsolve(A,B) solves the matrix equation AX = B, where A is a symbolic matrix and B is a symbolic column vector.

[X,R] = linsolve(A,B) also returns the reciprocal of the condition number of A if A is a square matrix. Otherwise, linsolve returns the rank of A.

Examples

Solve Linear Equations in Matrix Form

Solve this system of linear equations in matrix form by using linsolve.

$$\left[\begin{array}{ccc}2& 1& 1\\ -1& 1& -1\\ 1& 2& 3\end{array}\right]\left[\begin{array}{c}\mathit{x}\\ \mathit{y}\\ \mathit{z}\end{array}\right]=\left[\begin{array}{c}2\\ 3\\ -10\end{array}\right]$$

A = [ 2 1  1;
     -1 1 -1;
      1 2  3];
B = [2; 3; -10];
X = linsolve(A,B)

X = 3×1

     3
     1
    -5

The solutions are $\mathit{x}=3$, $\mathit{y}=1$, and $\mathit{z}=-5$.

Compute Condition Number of Square Matrix

Compute the reciprocal of the condition number of the square coefficient matrix by using two output arguments.

syms a x y z
A = [a 0 0; 0 a 0; 0 0 1];
B = [x; y; z];
[X, R] = linsolve(A, B)

X = 
$$\left(\begin{array}{c}\frac{x}{a}\\ \frac{y}{a}\\ z\end{array}\right)$$

R = 
$$\frac{1}{\max \left(\left|a\right|,1\right)\hspace{0.17em}\max \left(\frac{1}{\left|a\right|},1\right)}$$

Compute Rank of Nonsquare Matrix

If the coefficient matrix is rectangular, linsolve returns the rank of the coefficient matrix as the second output argument. Show this behavior.

syms a b x y
A = [a 0 1; 1 b 0];
B = [x; y];
[X,R] = linsolve(A,B)

Warning: Solution is not unique because the system is rank-deficient.

X = 
$$\left(\begin{array}{c}\frac{x}{a}\\ -\frac{x-a\hspace{0.17em}y}{a\hspace{0.17em}b}\\ 0\end{array}\right)$$

R = $$2$$

Input Arguments

A — Coefficient matrix

symbolic matrix

Coefficient matrix, specified as a symbolic matrix.

B — Right side of equations

symbolic vector | symbolic matrix

Right side of equations, specified as a symbolic vector or matrix.

Output Arguments

X — Solution

symbolic vector | symbolic matrix

Solution, returned as a symbolic vector or matrix.

R — Reciprocal condition number or rank

symbolic number | symbolic expression

Reciprocal condition number or rank, returned as a symbolic number of expression. If A is a square matrix, linsolve returns the condition number of A. Otherwise, linsolve returns the rank of A.

More About

Matrix Representation of System of Linear Equations

A system of linear equations is as follows.

$$\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\dots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\dots +a_{2n}{x}_n=b_2\\ \cdots \\ a_{m1}{x}_1+a_{m2}{x}_2+\dots +a_{mn}{x}_n=b_m\end{array}$$

This system can be represented as the matrix equation $$A\cdot \overrightarrow{x}=\overrightarrow{b}$$, where A is the coefficient matrix.

$$A=\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right)$$

$$\overrightarrow{b}$$ is the vector containing the right sides of equations.

$$\overrightarrow{b}=\left(\begin{array}{c}{b}_1\\ ⋮\\ b_m\end{array}\right)$$

Tips

• If the solution is not unique, linsolve issues a warning, chooses one solution, and returns it.

• If the system does not have a solution, linsolve issues a warning and returns X with all elements set to Inf.

• Calling linsolve for numeric matrices that are not symbolic objects invokes the MATLAB^® linsolve function. This function accepts real arguments only. If your system of equations uses complex numbers, use sym to convert at least one matrix to a symbolic matrix, and then call linsolve.

Version History

Introduced in R2012b