HW3

Problem SET 1

Question 1:- What is the rank of the matrix A?

Ans :- The rank of a matrix is defined as
(i) maximum number of linearly independent vectors columns in the matrix or
(ii) maximum number of linearly independent vectors row in the matrix. Are equal

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

4

## [1] 4

# Calculating rank of Matrix by using rankMatrix function of Matrix package.
require(Matrix)

## Loading required package: Matrix

rankMatrix(A)[1]

## [1] 4

Value is equal, hence Rank of matrix A = 4. It is full matrix and all 4 rows and columns are linearly independent to each other so Rank is 4.

Question 2:- Given an mxn matrix where m > n, what can be the maximum rank? The minimum rank, assuming that the matrix is non-zero?

Answer :- a) The maximum Rank for m * n where m>n is n if all n columns are linearly independent.
b) Minimum Rank for m *n where m >n is 1 as matrix is non-zero ; it will have one linear
independent columns.

Question 3:- What is the rank of matrix B?

Answer :- The RANK of matrix B is one as row 2 is equal to row1 multiply by 3 .Row 3 is equal to 2 times row 1. Therefore matrix has only one linear independent row. Rank is 1. In Next step we will calculate the rank of matrix B using rankMatrix function in R to prove it.

B <- matrix (c(1,3,2,2,6,4,1,3,2),nrow=3, ncol=3)
rankMatrix(B)[1]

## [1] 1

As shown above Rank of B is 1.

Problem SET 2

Question :-

Answer :- In the attached pdf I will share hand-written solution . Here we prove the same using r functions.

library(pracma)

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

## 
## Attaching package: 'pracma'

## The following objects are masked from 'package:Matrix':
## 
##     expm, lu, tril, triu

A <- matrix(c(1,0,0,2,4,0,3,5,6),nrow=3,ncol=3)
A

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

charpoly(A)

## [1]   1 -11  34 -24

Above values show that our Charcteristic polynomial is correct.

e <- eigen(A)
e$values

## [1] 6 4 1

Eigen values are correct.

e$vectors

##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0

Eigen vector values are same.