# mtimes, *

Symbolic matrix multiplication

## Description

example

A*B is the matrix product of A and B. If A is an m-by-p and B is a p-by-n matrix, then the result is an m-by-n matrix C defined as

$C\left(i,j\right)=\sum _{k=1}^{p}A\left(i,k\right)B\left(k,j\right)$

For nonscalar A and B, the number of columns of A must equal the number of rows of B. Matrix multiplication is not universally commutative for nonscalar inputs. That is, typically A*B is not equal to B*A. If at least one input is scalar, then A*B is equivalent to A.*B and is commutative.

mtimes(A,B) is equivalent to A*B.

## Examples

### Multiply Two Vectors

Create a 1-by-5 row vector and a 5-by-1 column vector.

syms x
A = [x, 2*x^2, 3*x^3, 4*x^4]
B = [1/x; 2/x^2; 3/x^3; 4/x^4]
A =
[ x, 2*x^2, 3*x^3, 4*x^4]

B =
1/x
2/x^2
3/x^3
4/x^4

Find the matrix product of these two vectors.

A*B
ans =
30

### Multiply Two Matrices

Create a 4-by-3 matrix and a 3-by-2 matrix.

A = sym('a%d%d', [4 3])
B = sym('b%d%d', [3 2])
A =
[ a11, a12, a13]
[ a21, a22, a23]
[ a31, a32, a33]
[ a41, a42, a43]

B =
[ b11, b12]
[ b21, b22]
[ b31, b32]

Multiply A by B.

A*B
ans =
[ a11*b11 + a12*b21 + a13*b31, a11*b12 + a12*b22 + a13*b32]
[ a21*b11 + a22*b21 + a23*b31, a21*b12 + a22*b22 + a23*b32]
[ a31*b11 + a32*b21 + a33*b31, a31*b12 + a32*b22 + a33*b32]
[ a41*b11 + a42*b21 + a43*b31, a41*b12 + a42*b22 + a43*b32]

### Multiply Matrix by Scalar

Create a 4-by-4 Hilbert matrix H.

H = sym(hilb(4))
H =
[   1, 1/2, 1/3, 1/4]
[ 1/2, 1/3, 1/4, 1/5]
[ 1/3, 1/4, 1/5, 1/6]
[ 1/4, 1/5, 1/6, 1/7]

Multiply H by eπ.

C = H*exp(sym(pi))
C =
[   exp(pi), exp(pi)/2, exp(pi)/3, exp(pi)/4]
[ exp(pi)/2, exp(pi)/3, exp(pi)/4, exp(pi)/5]
[ exp(pi)/3, exp(pi)/4, exp(pi)/5, exp(pi)/6]
[ exp(pi)/4, exp(pi)/5, exp(pi)/6, exp(pi)/7]

Use vpa and digits to approximate symbolic results with the required number of digits. For example, approximate it with five-digit accuracy.

old = digits(5);
vpa(C)
digits(old)
ans =
[ 23.141,  11.57, 7.7136, 5.7852]
[  11.57, 7.7136, 5.7852, 4.6281]
[ 7.7136, 5.7852, 4.6281, 3.8568]
[ 5.7852, 4.6281, 3.8568, 3.3058]

## Input Arguments

collapse all

Input, specified as a symbolic number, scalar variable, matrix variable (since R2021a), function, expression, or vector or matrix of symbolic scalar variables. Inputs A and B must be the same size unless one is a scalar. A scalar value expands into an array of the same size as the other input.

Input, specified as a symbolic number, scalar variable, matrix variable (since R2021a), function, expression, or vector or matrix of symbolic scalar variables. Inputs A and B must be the same size unless one is a scalar. A scalar value expands into an array of the same size as the other input.

## Version History

Introduced before R2006a