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Using cofactors of elements of second row, evaluate $$\Delta = \left| {\begin{array}{*{20}{c}} 5&3&8 \\ 2&0&1 \\ 1&2&3 \end{array}} \right|$$
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## QuestionMathsClass 12

Using cofactors of elements of second row, evaluate $$\Delta = \left| {\begin{array}{*{20}{c}} 5&3&8 \\ 2&0&1 \\ 1&2&3 \end{array}} \right|$$

Cofactor of $$a_{21}=A_{21}=\left(-1\right)^{2+1}\left|\begin{matrix}3&8\\2&3\end{matrix}\right|=\left(-1\right)^3\left(3\times 3-2\times 8\right)=-1\times \left(9-16\right)=-\left(-7\right)=7$$
Cofactor of
$$a_{22=}A_{22}=\left(-1\right)^{2+2}\left|\begin{matrix}5&8\\1&3\end{matrix}\right|=\left(-1\right)^4\left(5\times 3-1\times 8\right)=1\times \left(15-8\right)=7$$
Cofactor of  $$a_{23=}A_{23}=\left(-1\right)^{2+3}\left|\begin{matrix}5&3\\1&2\end{matrix}\right|=\left(-1\right)^5\left(5\times 2-1\times 3\right)=-1\times \left(10-3\right)=-7$$
Value of Determinant =$$\Delta = {a_{21}}{A_{21}} + {a_{22}}{A_{22}} + {a_{23}}{A_{23}}$$
$$=2\left(7\right)+0\left(7\right)+1\left(-7\right)$$
$$= 14 + 0 - 7$$
$$= 7$$