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,-2), nrow =4))
##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]   -1    0    1    3
## [3,]    0    1   -2    1
## [4,]    5    4   -2   -2
qr(A)$rank
## [1] 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.



The maximum rank of a matrix where m>n assuming non-zero matrix is n, the miminum rank will 1 as long as a matrix has at least 1 element

  1. What is the rank of matrix B?
(B<-matrix(c(1,3,2,
            2,6,4,
            1,3,2),nrow = 3))
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
qr(B)$rank
## [1] 1

Problem Set 2

Compute the eigenvalues and eigenvectors of the matrix A. You’ll need to show your work.

\[A = \begin{bmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 6\\ \end{bmatrix}\]

The rule of Sarros describes the characteristic equation for our given matrix as:

\[|A-\lambda I| = 0 = det \begin{bmatrix} 1-\lambda & 2 & 3\\ 0 & 4-\lambda & 5\\ 0 & 0 & 6-\lambda\\ \end{bmatrix}\]

meaning that our expected eigen values are 1, 4, and 6. We can compute the eigen vector as:

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

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