Exercise - T 10, page - 402

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.

Solution:

Let matrix \[A = \left[\begin{array} {rrr} -2 & 1 & -2 & -4 \\ 12 & 1 & 4 & 9 \\ 6 & 5 & -2 & -4 \\ 3 & -4 & 5 & 10 \end{array}\right] \]

#install.packages("pracma")
library(pracma)
A <- matrix(c(-2,1,-2,-4,12,1,4,9,6,5,-2,-4,3,-4,5,10), byrow = TRUE, nrow = 4)
# characteristic polynomial of A
chac_poly_A <- charpoly(A, info = FALSE)
chac_poly_A
## [1]   1  -7  18 -20   8
print("The constant term of the characteristic polynomial is 8")
## [1] "The constant term of the characteristic polynomial is 8"
# determinant of A
det_A <- det(A)
cat("The determinant of A is ", det_A)
## The determinant of A is  8

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