Question

Evaluate \(\begin{vmatrix}1&1&5 \\ 4&9&17 \\ 5&10&22\end{vmatrix}\).

• A. \(264\)
• B. \(1221\)
• C. \(0\)
• D. \(1\)

Hint:

Row-1 expansion uses \(+\,-\,+\). Linear dependence of rows \(\Rightarrow\) determinant \(0\).

Answer 1

The Correct Option is C

Idea. Use expansion along the first row (signs \(+\,-\,+\)). Compute three \(2\times2\) minors.
Step 1. Minors. \[ M_{11}=\begin{vmatrix}9&17 \\ 10&22\end{vmatrix}=9\cdot22-17\cdot10=198-170=28, \] \[ M_{12}=\begin{vmatrix}4&17 \\ 5&22\end{vmatrix}=4\cdot22-17\cdot5=88-85=3, \] \[ M_{13}=\begin{vmatrix}4&9 \\ 5&10\end{vmatrix}=4\cdot10-9\cdot5=40-45=-5. \]
Step 2. Combine with signs. \[ \det=1\cdot 28-1\cdot 3+5\cdot(-5)=28-3-25=0. \] Zero determinant also hints that the third row is the sum of the first two, so rows are dependent.