Exercise C12 problem

Exercise C12 solution

To solve this problem, I will utilize the charpoly (Characteristic Polynomial) function provided by R’s pracma (Practical Numerical Math Functions) package - https://www.rdocumentation.org/packages/pracma/versions/1.9.9/topics/charpoly.

# Create the 4 x 4 matrix A.
A <- matrix(c(1, 2, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 1, 0, 1), 4, 4)

# Find the characteristic polynomial of matrix A.
matrix_a_char_poly <- charpoly(A, info = TRUE)
## Error term: 4
# Output the result.
matrix_a_char_poly
## $cp
## [1]  1 -3 -2  2  2
## 
## $det
## [1] 2
## 
## $inv
##      [,1] [,2] [,3] [,4]
## [1,] -0.5  0.5  0.5    1
## [2,] -0.5 -0.5  1.5    2
## [3,]  1.0  0.0 -1.0   -3
## [4,]  0.0  0.0  0.0    1

Answer: The characteristic polynomial of matrix A is \(\lambda^4 -3\lambda^3 - 2\lambda^2 + 2\lambda + 2\)