Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
If $a, b, c$ are the roots of the equation $x^3 + 2x^2 + 1 = 0$, then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} =$
Step-by-Step Solution
Key Concept: Express the cyclic determinant in factored form and use given constraints to substitute known values systematically.
For the cyclic matrix $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$, the determinant equals $3abc - a^3 - b^3 - c^3 = -(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = 8$. Given constraints $a+b+c = -2$, $ab+bc+ca = 0$, and $abc = -1$, we compute $a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca) = 4$, yielding the determinant as $-(-2)(4-0) = 8$.
Correct Answer: 1