Evaluate the determinants: $(i)$ $\left|\begin{array}{ccc}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$ (ii) $\left|\begin{array}{ccc}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$ (iii) $\left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$ (iv) $\left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$
(A) $(i)$ Let $A = \left|\begin{array}{ccc}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$. Expanding along the second row: $|A| = -0\left|\begin{array}{cc}-1 & -2 \\ -5 & 0\end{array}\right| + 0\left|\begin{array}{cc}3 & -2 \\ 3 & 0\end{array}\right| - (-1)\left|\begin{array}{cc}3 & -1 \\ 3 & -5\end{array}\right| = 1(-15 - (-3)) = -15 + 3 = -12$.
(ii) Let $A = \left|\begin{array}{ccc}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$. Expanding along the first row: $|A| = 3(1 - (-6)) - (-4)(1 - (-4)) + 5(3 - 2) = 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46$.
(iii) Let $A = \left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$. Expanding along the first row: $|A| = 0(0 - (-9)) - 1(0 - 6) + 2(-3 - 0) = 0 + 6 - 6 = 0$.
(iv) Let $A = \left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$. Expanding along the first column: $|A| = 2(0 - 5) - 0 + 3(1 - (-4)) = 2(-5) + 3(5) = -10 + 15 = 5$.
(ii) Let $A = \left|\begin{array}{ccc}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$. Expanding along the first row: $|A| = 3(1 - (-6)) - (-4)(1 - (-4)) + 5(3 - 2) = 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46$.
(iii) Let $A = \left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$. Expanding along the first row: $|A| = 0(0 - (-9)) - 1(0 - 6) + 2(-3 - 0) = 0 + 6 - 6 = 0$.
(iv) Let $A = \left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$. Expanding along the first column: $|A| = 2(0 - 5) - 0 + 3(1 - (-4)) = 2(-5) + 3(5) = -10 + 15 = 5$.
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