Matrices & Determinants
Determinant Properties
Grade None
Question:
<p>If <i>x</i>, <i>y</i> and <i>z</i> are the integers in AP lying between 1 and 9 and <i>x</i>51, <i>y</i>41 and <i>z</i>31 are three digits numbers, the value of \begin{vmatrix} x & 51 \\ y & 41 \\ z & 31 \end{vmatrix} \div \begin{vmatrix} x & 5 \\ y & 4 \\ z & 3 \end{vmatrix} is</p>
<p>(a) <i>x</i> + <i>y</i> + <i>z</i></p>
<p>(b) <i>x</i> − <i>y</i> + <i>z</i></p>
<p>(c) 0</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: When rows of a determinant are linearly dependent (as in AP sequences), the determinant evaluates to zero.
<p>Since <i>x</i>, <i>y</i>, <i>z</i> are in AP and form a 3×2 matrix, the determinant property shows the first determinant equals 10 times the second, making their ratio yield 0 due to row linear dependence.</p>
Correct Answer: C