Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade Class 11
Question:
Let $a, z, y, z$ be real numbers satisfying the equations $az + ay = 5$, $x - ay = z$, $x + ay = az$, where $x, y, z$ are not all zero, then the number of the possible values of $a$ is
Step-by-Step Solution
Key Concept: A homogeneous system of linear equations has non-trivial solutions if and only if the coefficient determinant equals zero
For a homogeneous system $ax + by + cz = 0$ to have non-trivial solutions, the determinant must equal zero. Setting $D = \begin{vmatrix} a & a & -1 \\ 1 & a & -a \\ 1 & a & -a \end{vmatrix} = 0$, we expand: $a(a^2 - a) - a(a - (1)) + 1(0 - a) = 0 \Rightarrow a(a^2 - 2a + 1) = 0 \Rightarrow a = 0, 1$
<div class="key-concept"><strong>Key Concept:</strong> A homogeneous system of linear equations has non-trivial solutions if and only if the coefficient determinant equals zero</div>
<div class="trap-box"><strong>Trap:</strong> Students may forget that the condition for non-trivial solutions requires $D = 0$, not $D \neq 0$</div>
Correct Answer: 1