Let us find the characteristic polynomial p(λ) of the given matrix A=[
1
5
4
2
] p(λ)=|λ==I−==A| ⇒p(λ)=|
λ−1
5
4
λ−2
| ⇒p(λ)=(λ−1)(λ−2)−(4)(5) ⇒p(λ)=λ2−2λ−λ+2−20 ⇒p(λ)=λ2−3λ−18 According to the Cayley - Hamilton theorem, p(A)=0. ⇒A2−3A−18I=0 Now, the expression given to evaluate is A2−4A−12I, which can be written as: (A2−3A−18I)−A+6I =0−[