Inverse of Matrix in R

Operations on Lists in detail

The inverse of a matrix plays a crucial role in solving systems of linear equations. It is similar to the reciprocal of a number in regular arithmetic. If a matrix A has an inverse, denoted as A^-1, then their product results in an identity matrix:

A × A^-1 = I

Conditions for Matrix Inversion:

1. The matrix must be square (i.e., the number of rows equals the number of columns).
2. The determinant of the matrix must be non-zero (i.e., the matrix must be non-singular).

Methods to Compute the Inverse of a Matrix

There are two common ways to find the inverse of a matrix in R:

1. Using the solve() Function

The solve() function in R can be used to compute the inverse of a matrix. It can also be applied to solve linear equations of the form A_x=B.

Example:

# Define three vectors
v1 <- c(4, 3, 6)
v2 <- c(2, 5, 3)
v3 <- c(7, 1, 4)

# Combine them into a matrix
M <- rbind(v1, v2, v3)

# Print the original matrix
print(M)

# Compute the inverse using solve()
inv_M <- solve(M)

# Print the inverse matrix
print(inv_M)

Output:

[,1] [,2] [,3]
v1     4    3    6
v2     2    5    3
v3     7    1    4

               [,1]        [,2]       [,3]
[1,] -0.10714286  0.2142857  0.03571429
[2,]  0.10714286  0.0714286 -0.03571429
[3,]  0.21428571 -0.3571429  0.10714286

2. Using the inv() Function

The inv() function from the matlib package provides another way to compute the inverse of a matrix. Ensure that the matlib package is installed before using this function.

Example: Determinant of a Matrix

# Install and load matlib package (if not already installed)
install.packages("matlib")
library(matlib)

# Define three vectors
v1 <- c(2, 3, 7)
v2 <- c(5, 4, 2)
v3 <- c(8, 1, 6)

# Bind them into a matrix
M <- rbind(v1, v2, v3)

# Compute the determinant
print(det(M))

Output:

18

Example: Finding the Inverse Using inv()

# Compute the inverse using inv()
print(inv(t(M)))

Output:

[,1]       [,2]        [,3]
[1,] -0.05555556  0.3333333  0.11111111
[2,]  0.05555556  0.2222222 -0.11111111
[3,]  0.38888889 -0.4444444  0.05555556