Problem set 1

1. What is the rank of the matrix A?

\(A = \left( \begin{matrix} 1&2&3&4 \\ -1&0&1&3 \\ 0&1&-2&1 \\ 5&4&-2&-3 \end{matrix} \right)\)

A <- matrix(cbind(1,2,3,4,-1,0,1,3,0,1,-2,1,5,4,-2,-3), byrow = T, 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
# to calc matrix rank using rankMatrix function from Matrix pkg
require(Matrix)
## Loading required package: Matrix
rankMatrix(A)[1]
## [1] 4

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

For mxn matrix where m>n, the rank would be n if all n column are linearly independent. Assuming matrix is non-zero, the minimum rank would be 1 given at least 1 pivot.

3. What is the rank of matrix B?

\(B = \left( \begin{matrix} 1&2&1 \\ 3&6&3 \\ 2&4&2 \end{matrix} \right)\)

B <- matrix(cbind(1,2,1,3,6,3,2,4,2), byrow = T, ncol = 3)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2

Seeing the matrix B, row 2 is 3 times row 1 and row 3 is 2 times row 1 so we ended up having only one linear independent row therfore rank of B is 1.

here is using R.

rankMatrix(B)[1]
## [1] 1

Problem set 2

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.

\(A = \left( \begin{matrix} 1&2&3 \\ 0&4&5 \\ 0&0&6 \end{matrix} \right)\)

Link to solve this problem by hand https://github.com/amit-kapoor/data605/blob/master/607Assign3-Prob2-Manual.pdf

Please show your work using an R-markdown document.

library(pracma)
## 
## Attaching package: 'pracma'
## The following objects are masked from 'package:Matrix':
## 
##     expm, lu, tril, triu
A <- matrix(cbind(1,2,3,0,4,5,0,0,6), byrow = T, ncol = 3)
A
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
# to computes the characteristic polynomial
charpoly(A)
## [1]   1 -11  34 -24
eig <- eigen(A)

# eigen values of A
eig$values
## [1] 6 4 1
# eigen vectors of A
eig$vectors
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0