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.

Python3

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

Python3

# 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:

Python3

# 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:

Python3

# 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 be of dimension M*Z. 
Example:

Python3

# 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