1. Problem set 1
  1. What is the rank of the matrix A?

| 1 2 3 4| A =| −1 0 1 3| | 0 1 −2 1| | 5 4 −2 −3|

Convert A to RREF

| 1 2 3 4| RREF | 1 0 0 0| A =| −1 0 1 3| ——> | 0 1 0 0| | 0 1 −2 1| | 0 0 1 0| | 5 4 −2 −3| | 0 0 0 1|

rank is 4, there are no linear independent rows

A = matrix(c(1,2,3,4,-1,0,1,3,0,1,-2,1,5,4,-2,-3), nrow=4, byrow = TRUE)
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
pracma::rref(A)
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1
Matrix:: rankMatrix(A)
## [1] 4
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 8.881784e-16
  1. Given an mxn matrix where m > n, what can be the maximum rank? The minimum rank, assuming that the matrix is non-zero?

maximum rank would be m as it is possible as in problem set one that no rows are linearly independent. Tre minimum rank is 1

  1. What is the rank of matrix B?

      |1 2 1|   RREF    |1 2 1|

    B = |3 6 3| ——> |0 0 0| |2 4 2| |0 0 0|

B = matrix(c(1,2,1,3,6,3,2,4,2),nrow = 3, byrow = TRUE)
B
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    3    6    3
## [3,]    2    4    2
pracma::rref(B)
##      [,1] [,2] [,3]
## [1,]    1    2    1
## [2,]    0    0    0
## [3,]    0    0    0
Matrix::rankMatrix(B)
## [1] 1
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 6.661338e-16

####2 Problem set 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

     |1 2 3|
 A = |0 4 5|
     |0 0 6|
      | λ 0 0| 
 λI3 =| 0 λ 0| 
      | 0 0 λ|
       
           | λ-1 -2-3|
 λI3 = -A =| 0 λ -4-5| 
           | 0 0  λ-6|

apply rule of Sarrus

              | λ-1 -2 -3 | λ-1  -2
    λI3 = -A =| 0 λ -4 -5 |  0   λ-4 
              | 0 0  λ -6 |  0    0

(λ-1)(λ-4)(λ-6)+(-2)(-5)(0)+(-3)(0)(0)-(-2)(0)(λ-6)- -(λ-1)(-5)(0)-(-3)(λ-4)(0)

everything drops out except the first terms

     (λ-1)(λ-4)(λ-6)

this leaves us with three eingenvalues

          λ = 1
          λ = 4 
          λ = 6

eigenvector for
λ = 1

plug value in for lambda

        |0 -2 -3|   RREF   |0 1 0||v1|   |0|

λI3 = -A = |0 -3 -5| ——> |0 0 1||v2| = |0| |0 0 -5| |0 0 0||V3| |0| v1 = t v2 = 0 v3 = 0

               |1| 
        E1 = t |0|
               |0|

eigenvector for

         λ=4

plug value in for lambda

        |3 -2  -3|   RREF   |1  2/3  0||v1|   |0|

λI3 = -A = |0 0 -5| ——> |0 0 1||v2| = |0| |0 0 2| |0 0 0||v3| |0|

           v1 + 2/3v2 = 0
             v3 = 0

eigenvector for

         λ = 6

plug value in for lambda

        |5 -2 -3|  RREF    |1 0 -1.6||v1| |0|

λI3 = -A = |0 2 -5| ——-> |0 1 -2.5||v2|=|0| |0 0 0| |0 0 0 ||v3| |0|

        v1 - 1.6v3 = 0 -> 0.625v1 = v3
        v2 - 2.5v3 = 0 -> 0.4v2 = v3
        v3 = t
        
              |0.625|
        E6 = t|0.4  |
              |0    |
A = matrix(c(1,2,3,0,4,5,0,0,6), nrow = 3, byrow = TRUE)
A
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0    4    5
## [3,]    0    0    6
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
B = matrix(c(0,-2,-3,0,-3,-5,0,0,-5), nrow = 3, byrow = TRUE)
B
##      [,1] [,2] [,3]
## [1,]    0   -2   -3
## [2,]    0   -3   -5
## [3,]    0    0   -5
pracma::rref(B)
##      [,1] [,2] [,3]
## [1,]    0    1    0
## [2,]    0    0    1
## [3,]    0    0    0
C = matrix(c(3,-2,-3,0,0,-5,0,0,0), nrow =3, byrow = TRUE)
C
##      [,1] [,2] [,3]
## [1,]    3   -2   -3
## [2,]    0    0   -5
## [3,]    0    0    0
pracma::rref(C)
##      [,1]       [,2] [,3]
## [1,]    1 -0.6666667    0
## [2,]    0  0.0000000    1
## [3,]    0  0.0000000    0
D = matrix(c(5,-2,-3,0,2,-5,0,0,0), nrow = 3, byrow = TRUE)
D
##      [,1] [,2] [,3]
## [1,]    5   -2   -3
## [2,]    0    2   -5
## [3,]    0    0    0
pracma::rref(D)
##      [,1] [,2] [,3]
## [1,]    1    0 -1.6
## [2,]    0    1 -2.5
## [3,]    0    0  0.0