Operations on Matrices in R

Operations on Matrices in detail

Matrices in R consist of values, either real or complex numbers, arranged in a structured format with a fixed number of rows and columns. They are essential for representing data in an organized manner. Elements in a matrix should be enclosed in brackets or parentheses.

For example, a matrix with 9 elements is shown below:

[  1  2  3  ]
[  4  5  6  ]
[  7  8  9  ]

This matrix A has 3 rows and 3 columns. Each element can be identified by its row and column index. For instance, a[2,3] = 6.

The order of a matrix is determined by the number of rows and columns:

Order = Number of rows × Number of columns

So, matrix A has an order of 3 × 3.

Operations on Matrices

The fundamental matrix operations include Addition, Subtraction, Multiplication, and Division. Both matrices must have the same dimensions for these operations to be performed.

1. Matrix Addition: The sum of two matrices of the same order results in another matrix where each element is the sum of corresponding elements.

R Code for Matrix Addition

# Creating First Matrix
M1 = matrix(c(2, 4, 6, 8, 10, 12), nrow = 2, ncol = 3)

# Creating Second Matrix
M2 = matrix(c(1, 3, 5, 7, 9, 11), nrow = 2, ncol = 3)

# Performing Addition
result = M1 + M2

# Printing Result
print(result)

Output:

[,1] [,2] [,3]
[1,]    3    7   11
[2,]    5   11   17

2. Matrix Subtraction

Matrix subtraction works similarly, where each element of the second matrix is subtracted from the corresponding element of the first.

R Code for Matrix Subtraction

# Creating Matrices
M1 = matrix(c(5, 10, 15, 20, 25, 30), nrow = 2, ncol = 3)
M2 = matrix(c(2, 4, 6, 8, 10, 12), nrow = 2, ncol = 3)

# Performing Subtraction
result = M1 - M2

# Printing Result
print(result)

Output:

[,1] [,2] [,3]
[1,]    3    6    9
[2,]   12   15   18

3. Matrix Multiplication: Matrix multiplication involves multiplying corresponding elements when matrices have the same dimensions.

R Code for Matrix Multiplication

# Creating Matrices
M1 = matrix(c(2, 3, 4, 5, 6, 7), nrow = 2, ncol = 3)
M2 = matrix(c(1, 2, 3, 4, 5, 6), nrow = 2, ncol = 3)

# Element-wise Multiplication
result = M1 * M2

# Printing Result
print(result)

Output:

[,1] [,2] [,3]
[1,]    2    6   12
[2,]   20   30   42

4. Matrix Division: Each element of the first matrix is divided by the corresponding element of the second matrix.

R Code for Matrix Division

# Creating Matrices
M1 = matrix(c(10, 20, 30, 40, 50, 60), nrow = 2, ncol = 3)
M2 = matrix(c(2, 4, 6, 8, 10, 12), nrow = 2, ncol = 3)

# Element-wise Division
result = M1 / M2

# Printing Result
print(result)

Output:

[,1] [,2] [,3]
[1,]  5.0  5.0  5.0
[2,]  5.0  5.0  5.0

Properties of Matrix Operations

Matrix Addition Properties:

1. Commutative: M1 + M2 = M2 + M1
2. Associative: M1 + (M2 + M3) = (M1 + M2) + M3
3. Matrices must have the same dimensions.

Matrix Subtraction Properties:

1. Non-Commutative: M1 – M2 ≠ M2 – M1
2. Non-Associative: M1 – (M2 – M3) ≠ (M1 – M2) – M3
3. Matrices must have the same dimensions.

Matrix Multiplication Properties:

1. Commutative: M1 * M2 = M2 * M1
2. Associative: M1 * (M2 * M3) = (M1 * M2) * M3
3. Matrices must have the same dimensions.

Matrix Division Properties:

1. Non-Commutative: M1 / M2 ≠ M2 / M1
2. Non-Associative: M1 / (M2 / M3) ≠ (M1 / M2) / M3
3. Matrices must have the same dimensions.