Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_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 $A + 2B$ equals
Step-by-Step Solution
Key Concept: Column and row operations reduce a determinant to a factored form by creating zeros.
Perform column operations $C_1 \to C_1 + C_2 + C_3$ on the given determinant to factor out $(5x-4)$. Then use $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ to simplify. The determinant becomes $(5x-4)(x+4)^2 = (A+Bx)(x-A)^2$, giving $A=-4, B=5$, and $A+2B = -4+10 = 6$.
Correct Answer: 6