Matrices & Determinants
Determinant of polynomial in A using characteristic polynomial
MJAT_TS2_P2
Grade 12
Question:
Let $A$ be a $3\times 3$ matrix such that $\mathrm{tr}(A)=3$, $\mathrm{tr}(A^2)=5$, and $\mathrm{tr}(A^3)=15$. Find the value of $\det(A^2-3A+2I)$.
Step-by-Step Solution
Key Concept: Use Newton's identities: $c_1=\mathrm{tr}(A)=3$, $c_2=\frac{(c_1)^2-\mathrm{tr}(A^2)}{2}=\frac{9-5}{2}=2$. $\mathrm{tr}(A^3)=c_1\mathrm{tr}(A^2)-c_2\mathrm{tr}(A)+3c_3\Rightarrow 15=3\cdot 5-2\cdot 3+3c_3\Rightarrow c_3=\det(A)=2$. Characteristic polynomial: $\lambda^3-3\lambda^2+2\lambda-2=0$.
$\det(A)=2$. $\det(A^2-3A+2I)=\frac{8}{\det(A)}=\frac{8}{2}=\mathbf{4}$.
Correct Answer: 4