# mtimes, *

Matrix multiplication

## Syntax

``C = A*B``
``C = mtimes(A,B)``

## Description

example

````C = 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 `C` is an m-by-n matrix defined by $C\left(i,j\right)=\sum _{k=1}^{p}A\left(i,k\right)B\left(k,j\right).$This definition says that `C(i,j)` is the inner product of the `i`th row of `A` with the `j`th column of `B`. You can write this definition using the MATLAB® colon operator as C(i,j) = A(i,:)*B(:,j)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, `A*B` is typically not equal to `B*A`. If at least one input is scalar, then `A*B` is equivalent to `A.*B` and is commutative.```
````C = mtimes(A,B)` is an alternative way to execute `A*B`, but is rarely used. It enables operator overloading for classes.```

## Examples

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Create a 1-by-4 row vector, `A`, and a 4-by-1 column vector, `B`.

```A = [1 1 0 0]; B = [1; 2; 3; 4];```

Multiply `A` times `B`.

`C = A*B`
```C = 3 ```

The result is a 1-by-1 scalar, also called the dot product or inner product of the vectors `A` and `B`. Alternatively, you can calculate the dot product $A\cdot B$ with the syntax `dot(A,B)`.

Multiply `B` times `A`.

`C = B*A`
```C = 4×4 1 1 0 0 2 2 0 0 3 3 0 0 4 4 0 0 ```

The result is a 4-by-4 matrix, also called the outer product of the vectors `A` and `B`. The outer product of two vectors, $A\otimes B$, returns a matrix.

Create two arrays, `A` and `B`.

```A = [1 3 5; 2 4 7]; B = [-5 8 11; 3 9 21; 4 0 8];```

Calculate the product of `A` and `B`.

`C = A*B`
```C = 2×3 24 35 114 30 52 162 ```

Calculate the inner product of the second row of `A` and the third column of `B`.

`A(2,:)*B(:,3)`
```ans = 162 ```

This answer is the same as `C(2,3)`.

## Input Arguments

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Operands, specified as scalars, vectors, or matrices.

• `A` and `B` must be 2-D arrays, or one of them can be scalar.

• For nonscalar inputs, the number of columns in `A` must be equal to the number of rows in `B`.

• If one of `A` or `B` is an integer class (`int16`, `uint8`, …), then the other input must be a scalar. Operands with an integer data type cannot be complex.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `logical` | `char` | `duration` | `calendarDuration`
Complex Number Support: Yes

## Output Arguments

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Product, returned as a scalar, vector, or matrix. Array `C` has the same number of rows as input `A` and the same number of columns as input `B`. For example, if `A` is an m-by-0 empty matrix and `B` is a 0-by-n empty matrix, then `A*B` is an m-by-n matrix of zeros.

## Tips

• With chained matrix multiplications such as `A*B*C`, you might be able to improve execution time by using parentheses to dictate the order of the operations. Consider the case of multiplying three matrices with `A*B*C`, where `A` is 500-by-2, `B` is 2-by-500, and `C` is 500-by-2.

• With no parentheses, the order of operations is left to right so `A*B` is calculated first, which forms a 500-by-500 matrix. This matrix is then multiplied with `C` to arrive at the 500-by-2 result.

• If you instead specify `A*(B*C)`, then `B*C` is multiplied first, producing a 2-by-2 matrix. The small matrix then multiplies `A` to arrive at the same 500-by-2 result, but with fewer operations and less intermediate memory usage.