Data 605 Assignment 3

Libraries used:

library(knitr)

## Warning: package 'knitr' was built under R version 3.5.3

library(matrixcalc)
library(png)

Problem Sets

imgage <- "C:/Users/jpsim/Documents/Computationial Math/3.png"
include_graphics(imgage)

Problem Set 1

Solution 1.1:

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

##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]   -1    0    1    3
## [3,]    0    1   -2    1
## [4,]    5    4   -2   -3

matrix.rank(m)

## [1] 4

Solution 2.1:

# Maximum rank = $m$ (rows)    
# Minimum rank = 1 (all other rows could be linearly dependent)  

solution 1.3:

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

##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2

matrix.rank(m)

## [1] 1

Problem Set 2

Solution:

m <- matrix(data = c(1,0,0,2,4,0,3,5,6), nrow = 3, ncol = 3, byrow = FALSE)
m

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6

ev <- eigen(m)
ev <- ev$values
ev

## [1] 6 4 1