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

Answer

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 =\(\Delta = {a_{21}}{A_{21}} + {a_{22}}{A_{22}} + {a_{23}}{A_{23}}\)
\( =2\left(7\right)+0\left(7\right)+1\left(-7\right)\)
\( = 14 + 0 - 7\)
\(= 7\)
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