Assignment 3

Problem Set 1

#1

What is the rank of the matrix A?

[1 2 3 4]

[-1 0 1 3]

A=

[0 1 -2 1]

[5 4 -2 -3]

To solve this problem, I inserted matrix A into R and used the rankMatrix() function

#what is the rank of matrix A?

library(Matrix)
A=matrix(c(1,2,3,4,-1,0,1,3),nrow=4,ncol=4)

rank_A=rankMatrix(A)
rank_A
## [1] 2
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 8.881784e-16

Answer

The rank of matrix A is 2

#2

The maximum rank of a mxn matrix is n because rank is defined by the dimensions of column and row space. It cannot be more than n. The minimum rank of a non-zero matrix is 1 because it will at least have 1 row.

#3

What is the rank of matrix B?

[1 2 1]

B= [3 6 3 ]

[2 4 2]

To solve this problem, I inserted matrix B into R and again used the rankMatrix() function.

B=matrix(c(1,2,1,3,6,3,2,4,2),ncol=3, nrow=3)
rank_B=rankMatrix(B)
rank_B
## [1] 1
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 6.661338e-16

Answer

The rank of matrix B is 1.

Problem Set 2

Compute the convectors and eigenvalues of the given matrix.

For this question, I did the computation manually. Below is my work:

I will check my work with R below:

D=matrix(c(1,2,3,0,4,5,0,0,6), nrow=3, ncol=3)

eigen(D)
## eigen() decomposition
## $values
## [1] 6 4 1
## 
## $vectors
##      [,1]       [,2]        [,3]
## [1,]    0  0.0000000  0.83077316
## [2,]    0  0.3713907 -0.55384878
## [3,]    1 -0.9284767  0.05538488

My eigenvalues and eigenvectors match.