Problem 1

knitr::include_graphics('3-1.png')

1.1

# Reading library

library(matrixcalc)

# Feed the data into R

A <- matrix(c(1,-1,0,5,2,0,1,4,3,1,-2,-2,4,3,1,-3), nrow=4, ncol=4)
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
# Calculating rank of matrix with R

matrix.rank(A)
## [1] 4

1.2

Maximum rank for m x n where m > n is n whereas minimum rank for m x n matrix where m > n and having the assumption of non-zero is 1.

1.3

B <- matrix(c(1,3,2,2,6,4,1,3,2), nrow=3, ncol=3)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
#Rank for matrix B is:
matrix.rank(B)
## [1] 1

Problem 2

knitr::include_graphics('3-2.png')

# Creating matrix with the given numbers 

A2 <- matrix(c(1,0,0,2,4,0,3,5,6),nrow=3, ncol=3)
A2
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
# Using eigen function, we will print eigen and vector
result <- eigen(A2)

# Now, we will print out eigen and vectors from the saved value 'result'
result$values
## [1] 6 4 1
result$vectors
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0