Matrices & Determinants
Properties of Determinants
Grade None
Question:
<p>Let $D = \begin{vmatrix} a & a+d & a+3d \\ a+d & a+2d & a \\ a+2d & a & a+d \end{vmatrix}$, then:</p>
<p>(a) $D$ depends on $a$</p>
<p>(b) $D$ depends on $d$</p>
<p>(c) $D$ is independent of $a, d$</p>
<p>(d) $D = 0$</p>
Step-by-Step Solution
Key Concept: Check if rows or columns are linearly dependent; notice the symmetric pattern suggests $D=0$.
<p>Expand the determinant using cofactor expansion or row operations. Notice that the rows are linearly dependent or the determinant reduces to zero after simplification.</p>
Correct Answer: d