Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>If \(x, y, z\) are integers in AP, lying between 1 and 9, and \(x_{51}, y_{41}\) and \(z_{31}\) are three digit numbers, then the value of \(\begin{vmatrix}x_{51} & y_{41} & z_{31}\\x & y & z\\5 & 4 & 3\end{vmatrix}\) is</p>
<p>(a) \(x + y + z\)</p>
<p>(b) \(x - y + z\)</p>
<p>(c) 0</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: If two rows of a determinant have a linear dependency (one is a scalar multiple or linear combination of another), the determinant is zero.
<p><strong>Analysis:</strong> Since $x, y, z$ are in AP, we have $2y = x + z$, or $x - 2y + z = 0$. The three-digit numbers $x_{51}, y_{41}, z_{31}$ have a similar linear relationship: $x_{51} - 2y_{41} + z_{31} = 0$ (which can be verified by the structure of place values). Since Row 1 = Row 2 (in linear combination terms) and Row 3 = (5, 4, 3), the determinant equals 0.</p>
Correct Answer: C