# Matrix Multiplication in R

Matrix multiplication is the most useful matrix operation. It is widely used in areas such as network theory, transformation of coordinates and many more uses nowadays. A matrix in R can be created using `matrix()` function and this function takes input vector, nrow, ncol, byrow, dimnames as arguments.

Creating a matrix
A matrix can be created using matrix() function.

 `# R program to create a matrix ` `m <``-` `matrix(``1``:``8``, nrow``=``2``) ` `print``(m)  `

Output:

```     [,1] [,2] [,3] [,4]
[1,]    1    3    5    7
[2,]    2    4    6    8
```

#### Multiplication of Matrices

The multiplication operator * is used for multiplying a matrix by scalar or element-wise multiplication of two matrices.

Multilication with scalar

If you multiply a matrix with a scalar value, then every element of the matrix will be multiplied with that scalar.
Example:

 `# R program for matrix multiplication ` `# with a scalar ` `m <``-` `matrix(``1``:``8``, nrow``=``2``) ` `m <``-` `2``*``m    ` `print``(m) `

Output:

```     [,1] [,2] [,3] [,4]
[1,]    2    6   10   14
[2,]    4    8   12   16
```

In the above code, the scalar is multiplied with every element of the original matrix. This is how the multiplication process takes place:

```2*1=2  2*3=6  2*5=10  2*7=14
2*2=4  2*4=8  2*6=12  2*8=16
```

Multiplication between Matrices
When a matrix is multiplied with another matrix, the element-wise multiplication of two matrices take place. All the corresponding elements of both matrices will be multiplied under the condition that both matrices will be of the same dimension.
Example:

 `# R program for matrix multiplication ` ` `  `# Creating matrices ` `m <``-` `matrix(``1``:``8``, nrow``=``2``) ` `n <``-` `matrix(``8``:``15``, nrow``=``2``) ` ` `  `# Multiplying matrices ` `print``(m``*``n) `

Output:

```     [,1] [,2] [,3] [,4]
[1,]    8   30   60   98
[2,]   18   44   78  120
```

This is how the multiplication process takes place:

```1*8=8   3*10=30  5*12=60   7*14=98
2*9=18  4*11=44  6*13=78   8*15=120
```

Multiplication with Vector

If a matrix is multiplied with a vector then vector will be promoted to either row or column matrix to make two arguments conformable.
Example:

 `# R program for matrix multiplication ` ` `  `# Creating matrix ` `m <``-` `matrix(``1``:``8``, nrow``=``2``)    ` ` `  `# Creating a vector ` `vec <``-` `1``:``2`  ` `  `# Multiplying matrix with vector ` `print``(vec``*``m) `

Output:

```     [,1] [,2] [,3] [,4]
[1,]    1    3    5    7
[2,]    4    8   12   16
```

This is how the multiplication process takes place:

```
1*1=1   1*3=3   1*5=5   1*7=7
2*2=4   2*4=8   2*6=12  2*8=16
```

Multiplication using %*% operator

The Operator%*% is used for matrix multiplication satisfying the condition that the number of columns in the first matrix is equal to the number of rows in second. If matrix A[M, N] and matrix B[N, Z] are multiplied then the resultant matrix will of dimension M*N.
Example:

 `# R program for matrix multiplication ` ` `  `# Creating matrices ` `m <``-` `matrix(``1``:``8``, nrow``=``2``) ` `n <``-` `matrix(``8``:``15``, nrow``=``4``) ` ` `  `# Multiplying matrices using operator ` `print``(m ``%``*``%` `n) `

Output:

```     [,1] [,2]
[1,]  162  226
[2,]  200  280
```

This is how multiplication takes place:

```1*8+3*9+5*10+7*11 = 162      1*12+3*13+5*14+7*15=226
2*8+4*9+6*10+8*11 = 200      2*12+4*13+6*14+8*15=280
```

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