Question

Find the value of the determinant \(\begin{vmatrix}18&22&-13\\7&-9&11\\-11&7&17\end{vmatrix}\).

Hint:

Expand along the first row using 2x2 minors.

Answer 1

Step 1: Applying row reduction first to simplify:
Add Row1 + 2×Row3 to Row1's spirit isn't simpler here; instead expand directly along column 1 as an alternative cross-check: \(\Delta=18\begin{vmatrix}-9&11\\7&17\end{vmatrix}-7\begin{vmatrix}22&-13\\7&17\end{vmatrix}+(-11)\begin{vmatrix}22&-13\\-9&11\end{vmatrix}\).

Step 2: Computing these alternate minors:
\(\begin{vmatrix}-9&11\\7&17\end{vmatrix}=-230\). \(\begin{vmatrix}22&-13\\7&17\end{vmatrix}=374+91=465\). \(\begin{vmatrix}22&-13\\-9&11\end{vmatrix}=242-117=125\).

Step 3: Combining with correct cofactor signs:
\(\Delta=18(-230)-7(465)+(-11)(125)=-4140-3255-1375=-8770\), matching the first-row expansion.

Final Answer:
\[ \boxed{-8770} \]