1. The document describes an experiment using the R programming language to perform matrix operations. It involves creating matrices A and B, taking the inverse and transpose of A, multiplying A and B, estimating eigenvalues and eigenvectors of A, and finding the root of a function within an interval. 2. Key steps include using the matrix() function to create matrices A and B, using inv() and t() functions to take the inverse and transpose of A, multiplying the matrices with A * B, using the eigen() function to estimate eigenvalues and eigenvectors of A, and using the uniroot() function to find the root of a function between 0 and 8. 3. The document shows the code used to perform each

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LAB EXPERIMENT NO. 10 R Programming Language - Asst. Prof. Ashwini Mathur(SET-JAIN)
R Programming Language Lab
1. Use R-function matrix to create the matrices called A and B:
2. Take the inverse of A and the transpose of A.
3. Multiply A with B.
4. Estimate the eigenvalues and eigenvectors of A.
5. For a matrix A, x is an eigenvector, and λ the eigenvalue of a matrix A, if A • x = λ • x. Test it!
6. Create a new matrix, T, which equals P, except for the rst row, where the elements are 0.
7. Now estimate N= (I-T)-1, where I is the identity matrix.
8. Find the root of the equation f(x) = cos(2x)^3 in the interval [0, 8]. And draw the function curve.
Self work ......
9. Solve the equations 1000 = y ∗ (3 + x) ∗ (1 + y)4 for y and with x varying over the range from 1 to
Question:1. Use R-function matrix to create the matrices called A and B.
A
matrix:
3 × 3 of
type
dbl
1 1 -2
-1 2 1
0 1 -1
A
matrix:
3 × 3 of
type
dbl
6 2 0
8 -2 2
4 2 -2
A <- matrix( c(1,1,-2,-1,2,1,0,1,-1), nrow=3, byrow=TRUE)
B <- matrix( c(6, 2, 0,8,-2, 2,4, 2,-2), nrow=3, byrow=TRUE)
A #print matrix A
B #Print matrix B

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One of the most widely used kinds of matrix decomposition is called eigendecomposition, in which w
eigenvectors and eigenvalues.
Question:2.Take the inverse of A and the transpose of A.
Need to install the packgae and load the library for inverse of the matrix
Installing package into 'C:/Users/ashwinmathur/Documents/R/win-library/3.6'
(as 'lib' is unspecified)
Warning message:
"package 'matlib' is in use and will not be installed"
install.packages("matlib")
library(matlib)
det(A) != 0, so inverse exists Only non-singular matrices have an inverse
-2
-2
det(A) #Determinant of A
det(t(A))
A matrix: 3 ×
3 of type dbl
1.5 0.5 -2.5
0.5 0.5 -0.5
0.5 0.5 -1.5
AI = inv(A)
AI
A matrix: 3 ×
3 of type dbl
1.5 0.5 0.5
0.5 0.5 0.5
-2.5 -0.5 -1.5
transpose = t(AI) #Transpose of Matrix
transpose
2 · 1 · -1
e = eigen(A)
e$values
Question:3. Multiply A with B

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A
matrix:
3 × 3 of
type
dbl
6 2 0
-8 -4 2
0 2 2
Mul = A * B
Mul
Question:4.Estimate the eigenvalues and eigenvectors of A.
eigen() decomposition
$values
[1] 2 1 -1
$vectors
[,1] [,2] [,3]
[1,] 0.3015113 -0.8017837 7.071068e-01
[2,] 0.9045340 -0.5345225 -1.922963e-16
[3,] 0.3015113 -0.2672612 7.071068e-01
ev = eigen(A) #Eigen value of the matrix A
ev
x = ev$values #assign X to eigen values
lamda = ev$vectors #assign lamda to eigen vectors
Question:5. For a matrix A, x is an eigenvector, and λ the eigenvalue of a matrix A, if A • x = λ • x. Test
A matrix: 3 × 3 of type
lgl
FALSE FALSE FALSE
FALSE FALSE FALSE
FALSE FALSE FALSE
A*x == lamda*x
I <- matrix( c(1, 0, 0,0,1, 0,0, 0,1), nrow=3, byrow=TRUE)
A - (lamda*I)

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A matrix: 3 × 3 of type dbl
0.6984887 1.000000 -2.000000
-1.0000000 2.534522 1.000000
0.0000000 1.000000 -1.707107
A matrix: 3 × 3 of type dbl
0.6030227 -1.6035675 1.414214e+00
0.9045340 -0.5345225 -1.922963e-16
-0.3015113 0.2672612 -7.071068e-01
lamda*x
A
matrix:
3 × 3 of
type dbl
2 2 -4
-1 2 1
0 -1 1
A*x
Question:6. Create a new matrix, T, which equals P, except for the rst row, where the elements are 0.
T = matrix( c(0,0,0,-1,2,1,0,1,-1), nrow=3, byrow=TRUE)
P = matrix( c(1,1,-2,-1,2,1,0,1,-1), nrow=3, byrow=TRUE)
Question:7. Now estimate N= (I-T)-1, where I is the identity matrix.
A
matrix:
3 × 3
of type
dbl
1 0 0
0 1 0
0 0 1
I = matrix( c(1,0,0,0,1,0,0,0,1), nrow=3, byrow=TRUE)
I
Question:8. Now estimate N= (I-T)-1, where I is the identity matrix.
N = (I - T) - 1
N

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A
matrix:
3 × 3 of
type dbl
0 -1 -1
0 -2 -2
-1 -2 1
Question:9. Find the root of the equation f(x) = cos(2x)^3 in the interval [0, 8]. And draw the function c
fun = function (x){ cos(2*x)^3}
curve(fun(x), 0, 8)
abline(h = 0, lty = 3)
uni <- uniroot(fun, c(0, 8))$root
points(uni, 0, pch = 16, cex = 2)

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