#install.packages("pracma")
library(pracma)

Eigenvalues and Eigenvectors Page 402 Exercise T10 † Suppose that A is a square matrix. Prove that the constant term of the characteristic polynomial of A is equal to the determinant of A. # Make the matrix

A <- matrix(c(
    -2,6,12,9,
    4,9,11,10,
    7,12,0,2,
    31,6,1,3
), 4, 4)

A
##      [,1] [,2] [,3] [,4]
## [1,]   -2    4    7   31
## [2,]    6    9   12    6
## [3,]   12   11    0    1
## [4,]    9   10    2    3

Find constant of characterisitc polynomial

A_cp <- charpoly(A, info = TRUE)
## Error term: 7938
A_cp$cp
## [1]     1   -10  -578  -502 -3969

The constant term of the characteristic polynomial is -3969. # Find determinant

det(A)
## [1] -3969

From the above result we can prove that the constant term of the characteristic polynomial of A is equal to the determinant of A.