Evaluate the determinant: left| begin{array}{ | vedclass

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(A) To evaluate the determinant $\Delta = \left| \begin{array}{ccc} 1 & a & b \ -a & 1 & c \ -b & -c & 1 \end{array} ight|$,we expand along the fi

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$1 + a^2 + b^2 + c^2$

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(A) To evaluate the determinant $\Delta = \left| \begin{array}{ccc} 1 & a & b \ -a & 1 & c \ -b & -c & 1 \end{array} ight|$,we expand along the first row:$\Delta = 1 \cdot \left| \begin{array}{cc} 1 & c \ -c & 1 \end{array} ight| - a \cdot \left| \begin{array}{cc} -a & c \ -b & 1 \end{array} ight| + b \cdot \left| \begin{array}{cc} -a & 1 \ -b & -c \end{array} ight|$Calculating the $2 	imes 2$ determinants:$\Delta = 1(1 - (-c^2)) - a(-a - (-bc)) + b(ac - (-b))$$\Delta = 1(1 + c^2) - a(-a + bc) + b(ac + b)$$\Delta = 1 + c^2 + a^2 - abc + abc + b^2$$\Delta = 1 + a^2 + b^2 + c^2$.

Evaluate the determinant: $\left| \begin{array}{ccc} 1 & a & b \\ -a & 1 & c \\ -b & -c & 1 \end{array} \right|$

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