Question:
Evaluate \(\begin{vmatrix}1&2&-1 \\ 5&4&1 \\ 7&6&1\end{vmatrix}\).
Evaluate \(\begin{vmatrix}1&2&-1 \\ 5&4&1 \\ 7&6&1\end{vmatrix}\).
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If one row = sum of two others, det \(=0\). Spotting such relations saves time.
Updated On: Oct 17, 2025
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The Correct Option is A
Solution and Explanation
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). \[ \det=1\!\begin{vmatrix}4&1 \\ 6&1\end{vmatrix}-2\!\begin{vmatrix}5&1 \\ 7&1\end{vmatrix}+(-1)\!\begin{vmatrix}5&4 \\ 7&6\end{vmatrix} =1(-2)-2(-2)+(-1)(2)=-2+4-2=0. \]
Step (verification). \[ \det=1\!\begin{vmatrix}4&1 \\ 6&1\end{vmatrix}-2\!\begin{vmatrix}5&1 \\ 7&1\end{vmatrix}+(-1)\!\begin{vmatrix}5&4 \\ 7&6\end{vmatrix} =1(-2)-2(-2)+(-1)(2)=-2+4-2=0. \]
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