Problem Set 1

What is the rank of the matrix A?

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

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

Rank:

library(Matrix)
library(pracma)
## 
## Attaching package: 'pracma'
## The following objects are masked from 'package:Matrix':
## 
##     expm, lu, tril, triu
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
rankMatrix(A)[1]
## [1] 4

Rank(A) = 4

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

assuming m rows and n columns then the maximum rank would be the minimum between m or n. (ie 4x5 matrix -> 4 5x2 matrix -> 2)

The minimum rank would be 1 for a non-zero matrix

  1. What is the rank of matrix B?

\[ B = \left[ {\begin{array}{cc} 1 & 2 & 1\\ 3 & 6 & 3\\ 2 & 4 & 2\\ \end{array}} \right] \]

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
rref(B)
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    0    0    0
## [3,]    0    0    0
rankMatrix(B)[1]
## [1] 1

We can see 1 non-zero row in the reduced row echelon form of matrix B thus the rank is 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.

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

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
eigen(A)$values
## [1] 6 4 1
eigen(A)$vectors
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0
charpoly(A)
## [1]   1 -11  34 -24