Question:
Find the value of the determinant \(\begin{vmatrix}18&22&-13\\7&-9&11\\-11&7&17\end{vmatrix}\).
Find the value of the determinant \(\begin{vmatrix}18&22&-13\\7&-9&11\\-11&7&17\end{vmatrix}\).
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Expand along the first row using 2x2 minors.
Updated On: Sep 23, 2026
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Solution and Explanation
Step 1: Key Formula — expansion along the first row:
\(\Delta=18\begin{vmatrix}-9&11\\7&17\end{vmatrix}-22\begin{vmatrix}7&11\\-11&17\end{vmatrix}+(-13)\begin{vmatrix}7&-9\\-11&7\end{vmatrix}\).
Step 2: Computing the 2x2 minors:
\(\begin{vmatrix}-9&11\\7&17\end{vmatrix}=(-9)(17)-(11)(7)=-153-77=-230\). \(\begin{vmatrix}7&11\\-11&17\end{vmatrix}=(7)(17)-(11)(-11)=119+121=240\). \(\begin{vmatrix}7&-9\\-11&7\end{vmatrix}=(7)(7)-(-9)(-11)=49-99=-50\).
Step 3: Substituting back:
\(\Delta=18(-230)-22(240)+(-13)(-50)=-4140-5280+650\).
Final Answer:
\(\Delta=-4140-5280+650=-8770\).\[ \boxed{-8770} \]
\(\Delta=18\begin{vmatrix}-9&11\\7&17\end{vmatrix}-22\begin{vmatrix}7&11\\-11&17\end{vmatrix}+(-13)\begin{vmatrix}7&-9\\-11&7\end{vmatrix}\).
Step 2: Computing the 2x2 minors:
\(\begin{vmatrix}-9&11\\7&17\end{vmatrix}=(-9)(17)-(11)(7)=-153-77=-230\). \(\begin{vmatrix}7&11\\-11&17\end{vmatrix}=(7)(17)-(11)(-11)=119+121=240\). \(\begin{vmatrix}7&-9\\-11&7\end{vmatrix}=(7)(7)-(-9)(-11)=49-99=-50\).
Step 3: Substituting back:
\(\Delta=18(-230)-22(240)+(-13)(-50)=-4140-5280+650\).
Final Answer:
\(\Delta=-4140-5280+650=-8770\).\[ \boxed{-8770} \]
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