Determinants
General
Grade 12
Question:
<p>If the system of linear equations, $x + 2ay + az = 0, x + 3by + bz = 0$ and $x + 4cy + cz = 0$ has a non-zero solution, then $a, b, c$ satisfy</p>
2ac = ab + bc
2ab = ac + bc
2b = a + c
b^2 = ac
Step-by-Step Solution
Key Concept: General
For a non-zero solution, the determinant of the coefficients must be zero: $$\Delta = \begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{vmatrix} = 0$$ Applying row operations $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_1$: $$\begin{vmatrix} 1 & 2a & a \\ 0 & 3b-2a & b-a \\ 0 & 4c-2a & c-a \end{vmatrix} = 0$$ $$\Rightarrow (3b-2a)(c-a) - (4c-2a)(b-a) = 0$$ $$\Rightarrow 3bc - 3ab - 2ac + 2a^2 - (4bc - 4ac - 2ab + 2a^2) = 0$$ $$\Rightarrow -bc - ab + 2ac = 0 \Rightarrow 2ac = ab + bc$$
Correct Answer: A