Evaluate the determinant | ccc 3 & -1 & -2 0 & 0 & -1 3 & -5 & 0 … - Vidyadip

Question

Evaluate the determinant $\left|\begin{array}{ccc}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$

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Answer

Given: $\left| {\begin{array}{*{20}{c}} 3&{ - 1}&{ - 2} \\ 0&0&{ - 1} \\ 3&{ - 5}&0 \end{array}} \right|$
Expanding along first row, $3\left| {\begin{array}{*{20}{c}} 0&{ - 1} \\ { - 5}&0 \end{array}} \right| - \left( { - 1} \right)\left| {\begin{array}{*{20}{c}} 0&{ - 1} \\ 3&0 \end{array}} \right| + \left( { - 2} \right)\left| {\begin{array}{*{20}{c}} 0&0 \\ 3&{ - 5} \end{array}} \right|$
= 3(0 - 5) + 1(0 - (-3)) - 2(0 - 0)
= -15 + 3 - 0 = -12

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