Question

Evaluate \(\begin{vmatrix}1&2&-1 \\ 5&4&1 \\ 7&6&1\end{vmatrix}\).

• A. \(0\)
• B. \(1\)
• C. \(-1\)
• D. \(12\)

Hint:

If one row = sum of two others, det \(=0\). Spotting such relations saves time.

Answer 1

The Correct Option is A

Idea. Again expand along the first row, or notice a quick pattern: \(R_3=R_1+R_2\) \(\Rightarrow\) rows dependent \(\Rightarrow\) determinant \(0\). We also verify by expansion.
Step (verification). \[ \det=1\!\begin{vmatrix}4&1 \\ 6&1\end{vmatrix}-2\!\begin{vmatrix}5&1 \\ 7&1\end{vmatrix}+(-1)\!\begin{vmatrix}5&4 \\ 7&6\end{vmatrix} =1(-2)-2(-2)+(-1)(2)=-2+4-2=0. \]