1 Problem Set 1

\[\mathbf{A} = \left[\begin{array} {rrr} 1 & 2 & 3 & 4 \\ -1 & 0 & 1 & 3 \\ 0 & 1 & 2 & 1 \\ 5 & 4 & -2 & -3 \end{array}\right]\]

1.1 What is the rank of the matrix A?

#Also, tried some Quick r functions (https://www.statmethods.net/advstats/matrix.html)
#QR decomposition of A. 
# Finding the rank of A
A <- matrix(c(1,2,3,4,
              -1,0,1,3,
              0,1,-2,1,
              5,4,-2,-3), 4, byrow=T)

print(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
#rank of A
qr(A)$rank
## [1] 4

Also, using pracma library

library(pracma)
rref(A)
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1
Rank(A)
## [1] 4

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

If m is grater than n, the maximum rank is n. Assuming that the matrix has at least one non-xero element, its minimum rank should be 1.

1.3 What is the rank of matrix B?

\[\mathbf{A} = \left[\begin{array} {rrr} 1 & 2 & 1 \\ 3 & 6 & 3 \\ 2 & 4 & 2 \end{array}\right]\]

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

print(B)
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
#Ranks of B
qr(B)$rank
## [1] 1

Also, using pracma library

rref(B)
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    0    0    0
## [3,]    0    0    0
#Ranks of B
Rank(B)
## [1] 1

2 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.

\[\mathbf{A} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{array}\right]\]

To determine the eigenvalues of \[\lambda \space\ of\space\ A \], we initially solve for the determinant of \[A-\lambda*I=0\]

\[(1-\lambda)[(4-\lambda)(6-\lambda)-5×0]+2[0×(6-\lambda)-5×0]+3[0×0-0(6-\lambda)]=0\]

The eigenvalues are: \[(\lambda-1)(\lambda-4)(\lambda-6)=0\]

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

print(A)
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
eigen(A)$values
## [1] 6 4 1
charpoly(A)
## [1]   1 -11  34 -24

Reference: https://www.statmethods.net/advstats/matrix.html