Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2$, then the ordered pair $(A, B)$ is equal to
(4, 5)
(-4, -5)
(-4, 3)
( -4, 5)
Step-by-Step Solution
Key Concept: Determinants can be expressed as polynomial functions, and specific values give constraints on parameters.
Setting $z = 0$ in the given determinant $\begin{vmatrix} z-1 & 2x & 2x \\ 2x & z-4 & 2x \\ 2x & 2x & z-4 \end{vmatrix} = (A+Bz)(z-A)^2$, we get $\begin{vmatrix} -1 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & -4 \end{vmatrix} = A^3 = -4$, confirming $A = -4$. Further analysis with $z \to \infty$ yields $B = 5$, and the characteristic equation gives the final answer as $-4$.
Correct Answer: -4