Problem Set 1-1

Ques: What is the rank of the matrix A?

# Creating Matrix A

A <-matrix(c(1,2,3,4,-1,0,1,3,0,1,-2,1,5,4,-2,-3), nrow=4,byrow=TRUE)

#displaying the Matrix 
A
##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]   -1    0    1    3
## [3,]    0    1   -2    1
## [4,]    5    4   -2   -3
#Determining the rank of A
qr(A)$rank
## [1] 4

Ans: Therefore the Rank of Matrix A is 4

Problem Set 1-2

Ques: Given an mxn matrix where m > n, what can be the maximum rank? The minimum rank, assuming that the matrix is non-zero?

Ans: Given a mxn matrix with m>n the min rank is 1 and the maximum rank is n assuming that this is a non zero matrix.

Problem Set 1-3

Ques: What is the rank of matrix B?

# Creating Matrix B

B <- matrix(c(1,2,1,3,6,3,2,4,2), nrow=3, byrow=TRUE)

#displaying the Matrix 
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
#Determining the rank of A
qr(B)$rank
## [1] 1

Ans: Therefore the Rank of Matrix B is 1

Problem Set 2-1

Ques: Compute the eigenvalues and eigenvectors of the matrix A. You’ll need to show your work. You’ll need to write out the characteristic polynomial and show your solution.

Ans: Solving for eigen values and Characteristic polynomial by hand

Testing the answers through R

#Matrix A
A <- matrix(c(1,2,3,0,4,5,0,0,6), nrow=3, byrow=TRUE)

#Displaying Matrix A
A
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
eigen(A)
## eigen() decomposition
## $values
## [1] 6 4 1
## 
## $vectors
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0
library(pracma)

# Characteristic Polynomial 
charpoly(A)
## [1]   1 -11  34 -24