Problem Set 1

(1) What is the rank of the Matrix A?

# Define matrix A.
A <- matrix(c(1, 2, 3, 4, -1, 0, 1, 3, 0, 1, -2, 1, 5, 4, -2, -3), 4, 4, byrow = TRUE)

# Calculate the Rank of matrix A.
matrix_rank <- rankMatrix(A)[1]

# Output the result.
matrix_rank
## [1] 4

Answer: The rank of matrix A is 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?

Answer: Given the fact that \(m > n\), then the maximum rank is \(n\). If the matrix contains at least one non-zero element, then its minimum rank is 1.

(3) What is the rank of matrix B?

# Define matrix B.
B <- matrix(c(1, 2, 1, 3, 6, 3, 2, 4, 2), 3, 3, byrow = TRUE)

# Calculate the Rank of matrix B.
matrix_rank <- rankMatrix(B)[1]

# Output the result.
matrix_rank
## [1] 1

Answer: The rank of martix B is 1.

Problem Set 2

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.

Characteristic Polynomial.
# Define matrix A.
A <- matrix(c( 1, 2, 3, 0, 4, 5, 0, 0, 6), 3, 3, byrow = TRUE)

# Find the characteristic polynomial of matrix A.
matrix_a_char_poly <- charpoly(A, info = TRUE)
## Error term: 0
# Output the result.
matrix_a_char_poly
## $cp
## [1]   1 -11  34 -24
## 
## $det
## [1] 24
## 
## $inv
##      [,1]  [,2]        [,3]
## [1,]    1 -0.50 -0.08333333
## [2,]    0  0.25 -0.20833333
## [3,]    0  0.00  0.16666667

Answer: The characteristic polynomial of matrix A is \(\lambda^3 - 11\lambda^2 + 34\lambda - 24 = 0\).

Eigenvalues.
# Get the Eigenvalues of Matrix A.
eigen <- eigen(A)
eigen_values <- eigen$values

# Output the result.
eigen_values
## [1] 6 4 1

Answer: The eigenvalues of matrix A are 6, 4, and 1.

Eigenvectors.

# Get the Eigenvectors of Matrix A.
eigen_vectors <- eigen$vectors

# Output the result.
eigen_vectors
##           [,1]      [,2] [,3]
## [1,] 0.5108407 0.5547002    1
## [2,] 0.7981886 0.8320503    0
## [3,] 0.3192754 0.0000000    0

Answer: The eigenvectors of matrix A are as follows:

\(\lambda = 6\),

\(\left[\begin{array}{}0.5108407 \\0.7981886 \\0.3192754\end{array}\right]\)

\(\lambda = 4\),

\(\left[\begin{array}{}0.5547002 \\0.8320503 \\0\end{array}\right]\)

\(\lambda = 1\),

\(\left[\begin{array}{}1 \\0 \\0\end{array}\right]\)