Problem set 1

1. What is the rank of the matrix A?

library("Matrix")
## Warning: package 'Matrix' was built under R version 3.5.3
A = matrix(c(1,-1,0,5,2,0,1,4,3,1,-2,-2,4,3,1,-3),4,4)
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
rankMatrix(A)[1]
## [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?

The maximum rank a matrix mxn where m>n is min(m,n) which is n. The minimum rank of any non-zero matrix is 1.

3. What is the rank of matrix B?

B = matrix(c(1,3,2,2,6,4,1,3,2),3,3)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
rankMatrix(B)[1]
## [1] 1

Problem set 2

1. 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.

\[\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 0 \end{bmatrix}\] \[\begin{bmatrix} 1-\lambda & 2 & 3 \\ 0 & 4-\lambda & 5 \\ 0 & 0 & 0-\lambda \end{bmatrix}\]

Calculating the determinant of the above matrix, we get the Eigen values as 1,4,6.
The process of getting the Eigen values and corresponding Eigen vectors is shown in the images below.

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