HW 3

Problem Set 1

library(Matrix)
library(pracma)

1

A <- matrix(c(1,-1,0,5,2,0,1,4,3,1,-2,-2,4,3,1,-3),nrow= 4)
det(A)
## [1] -9
rankMatrix(A,warn.t = FALSE)[1]
## [1] 4

The determinant of this matrix is not zero so the rank is 4. We can also see that the rank is 4 by using the rankMatrix function.

2

C <- matrix(c(1,2,3,4,3,5),ncol=2,nrow=3)
C
##      [,1] [,2]
## [1,]    1    4
## [2,]    2    3
## [3,]    3    5
rankMatrix(C,warn.t = FALSE)[1]
## [1] 2

The maximum rank for a mxn matrix with m > n would be n since the rank cannot be greater than the smaller column and/or row. THe minimum rank rank would be 1 given the case that matrix is non-zero.

3

B <- matrix(c(1,3,2,2,6,4,1,3,2),nrow=3)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
det(B)
## [1] 0
rankMatrix(B)[1]
## [1] 1

The determinant of this matrix is zero so the rank is between 1 and 2. We see that 2(row1) = row3 and 3(row1) - row3 so the matrix has row1 as the only independet row making the rank=1. We can also see that the rank is 1 by using the rankMatrix function.

Problem Set 2

Characteristic Polynomial:
\[\mathbf{det(A - xI)} =det(\left[\begin{array} {rrr} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 6\\ \end{array}\right] - x \left[\begin{array} {rrr} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{array}\right] ) \]
\[\mathbf{det(A - xI)} =det(\left[\begin{array} {rrr} 1-x & 2 & 3\\ 0 & 4-x & 5\\ 0 & 0 & 6-x\\ \end{array}\right]) \] \[-x^3 + 11x^2 - 34x + 24\]
Eigenvalues:

polyroot(c(24,-34,11,-1))
## [1] 1+0i 4-0i 6+0i

Eigenvalues are 1, 4, and 6.

Eigenvectors:

A <-  matrix(c(1,0,0,2,4,0,3,5,6),nrow=3)
A
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
diag(3)
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1

Eigenvector for 1:

rref(A-1*diag(3))
##      [,1] [,2] [,3]
## [1,]    0    1    0
## [2,]    0    0    1
## [3,]    0    0    0

\[x_2=0\quad x_3=0\quad x_1=1\] \[\mathbf{E_1} =\left[\begin{array} {rrr} 1\\ 0\\ 0\\ \end{array}\right] \] Eigenvector for 4:

rref(A-4*diag(3))
##      [,1]       [,2] [,3]
## [1,]    1 -0.6666667    0
## [2,]    0  0.0000000    1
## [3,]    0  0.0000000    0

\[x_1 - .67(x_2)=0\quad x_3=0\] \[1-.67(x_2) = 0\] \[x_2 = 1.5\]
\[\mathbf{E_4} =\left[\begin{array} {rrr} 1\\ 1.5\\ 0\\ \end{array}\right] \] Eigenvector for 6:

rref(A-6*diag(3))
##      [,1] [,2] [,3]
## [1,]    1    0 -1.6
## [2,]    0    1 -2.5
## [3,]    0    0  0.0

\[x_1 - 1.6(x_3)=0\quad x_2 - 2.5(x_3)=0\] \[\mathbf{E_6} =\left[\begin{array} {rrr} 1.6\\ 2.5\\ 1\\ \end{array}\right] \]