Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
If $$\begin{vmatrix} x & x+y & x+y+z \\ 2x & 3x+2y & 4x+3y+2z \\ 3x & 6x+3y & 10x+6y+3z \end{vmatrix} = -64$$, then the real value of $x$ is ____
Step-by-Step Solution
Key Concept: Use column operations to factor out x from the determinant. Apply C₃ → C₃ - 2C₁ and C₂ → C₂ - xC₁ to simplify the matrix into a form where x³ can be factored out, reducing the problem to solving x³ = 64.
Express the given $3 \times 3$ determinant in terms of $x$ and perform column operations $C_3 \to C_3 - 2C_1$ and $C_2 \to C_2 - xC_1$. The resulting determinant becomes $x^3(6-8+3) = 64$, so $x^3 = 64$ and $x = 4$.
Correct Answer: 4