Question:
Find the value of the determinant \(\begin{vmatrix}17&15&13\\9&-8&7\\-3&2&5\end{vmatrix}\).
Find the value of the determinant \(\begin{vmatrix}17&15&13\\9&-8&7\\-3&2&5\end{vmatrix}\).
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Expand the 3×3 determinant along the first row using cofactors.
Updated On: Sep 23, 2026
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Solution and Explanation
Step 1: Expanding along the first row:
\(\Delta=17\begin{vmatrix}-8&7\\2&5\end{vmatrix}-15\begin{vmatrix}9&7\\-3&5\end{vmatrix}+13\begin{vmatrix}9&-8\\-3&2\end{vmatrix}\).
Step 2: Evaluating the 2×2 minors:
\(\begin{vmatrix}-8&7\\2&5\end{vmatrix}=-40-14=-54\); \(\begin{vmatrix}9&7\\-3&5\end{vmatrix}=45+21=66\); \(\begin{vmatrix}9&-8\\-3&2\end{vmatrix}=18-24=-6\).
Step 3: Combining:
\(\Delta=17(-54)-15(66)+13(-6)=-918-990-78\).
Final Answer:
\[ \boxed{\Delta=-1986} \]
\(\Delta=17\begin{vmatrix}-8&7\\2&5\end{vmatrix}-15\begin{vmatrix}9&7\\-3&5\end{vmatrix}+13\begin{vmatrix}9&-8\\-3&2\end{vmatrix}\).
Step 2: Evaluating the 2×2 minors:
\(\begin{vmatrix}-8&7\\2&5\end{vmatrix}=-40-14=-54\); \(\begin{vmatrix}9&7\\-3&5\end{vmatrix}=45+21=66\); \(\begin{vmatrix}9&-8\\-3&2\end{vmatrix}=18-24=-6\).
Step 3: Combining:
\(\Delta=17(-54)-15(66)+13(-6)=-918-990-78\).
Final Answer:
\[ \boxed{\Delta=-1986} \]
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