1. Problem set 1

(1) What is the rank of the matrix A?

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
qr(a)
## $qr
##            [,1]       [,2]      [,3]      [,4]
## [1,] -5.1961524 -4.2339020  1.539601  2.694301
## [2,] -0.1924501 -1.7533038 -1.436442 -4.795180
## [3,]  0.0000000  0.5703518  3.683241  2.162187
## [4,]  0.9622504 -0.5877259  0.592032  0.268209
## 
## $rank
## [1] 4
## 
## $qraux
## [1] 1.192450 1.573827 1.805914 0.268209
## 
## $pivot
## [1] 1 2 3 4
## 
## attr(,"class")
## [1] "qr"
qr(a)$rank
## [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?

If m is > n, then the maximum rank is n. if n > m then the maximum rank is m.

(3) What is the rank of matrix B?

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
qr(b)
## $qr
##            [,1]       [,2]          [,3]
## [1,] -3.7416574  2.1380899  1.069045e+00
## [2,] -0.8017837 -7.1713717 -3.585686e+00
## [3,]  0.5345225  0.5658953 -4.440892e-16
## 
## $rank
## [1] 2
## 
## $qraux
## [1] 1.267261e+00 1.824477e+00 4.440892e-16
## 
## $pivot
## [1] 1 2 3
## 
## attr(,"class")
## [1] "qr"
qr(b)$rank
## [1] 2

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.

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

\(\leftthreetimes\) is an eigenvalue of A

\(\left[ \begin{matrix} \leftthreetimes & 0 & 0 \\ 0 & \leftthreetimes & 0 \\ 0 & 0 & \leftthreetimes \end{matrix} \right]\) - \(\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{matrix} \right]\) \(\left[ \begin{matrix} \leftthreetimes -1 & -2 & -3 \\ 0 & \leftthreetimes -4 & -5 \\ 0 & 0 & \leftthreetimes -6 \end{matrix} \right]\)

$$

\(((\leftthreetimes -1)(\leftthreetimes -4)(\leftthreetimes -6))+(-2*-5*0)+(-3*0*0)-\\ (-2*0*(\leftthreetimes -6))-((\leftthreetimes -1)*-5*0)-(-3*(\leftthreetimes -4)*0)\)

all the multiplications with 0’s cancel out and youre left with:

\(((\leftthreetimes -1)(\leftthreetimes -4)(\leftthreetimes -6))\)

\(((\leftthreetimes -1)(\leftthreetimes -4)(\leftthreetimes -6))=0\\ ({ \leftthreetimes }^{ 2 }-5\leftthreetimes +4)(\leftthreetimes -6)=0\\ ({ \leftthreetimes }^{ 3 }-5{ \leftthreetimes }^{ 2 }+4\leftthreetimes -6{ \leftthreetimes }^{ 2 }+30\leftthreetimes -24)=0\\ ({ \leftthreetimes }^{ 3 }-11{ \leftthreetimes }^{ 2 }+34\leftthreetimes -24)=0\)

\(\leftthreetimes =1,\leftthreetimes =6,\leftthreetimes =4\)

\(\leftthreetimes =1\) $$ $$

\(\leftthreetimes =6\) $$ $$

\(\leftthreetimes =4\) $$ $$

print("end")
## [1] "end"