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Solve linear equations in matrix form


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



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


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


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

From X, x = 3, y = 1 and z = –5.

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 =
R =
1/(max(abs(a), 1)*max(1/abs(a), 1))

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.
  In sym.linsolve at 67 
X =
 -(x - a*y)/(a*b)
R =

Input Arguments

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Coefficient matrix, specified as a symbolic matrix.

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

Output Arguments

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Solution, returned as a symbolic vector or matrix.

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

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Matrix Representation of System of Linear Equations

A system of linear equations is as follows.


This system can be represented as the matrix equation Ax=b, where A is the coefficient matrix.


b is the vector containing the right sides of equations.



  • 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.

Introduced in R2012b