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Write minors and cofactors of the element of \(\left| {\begin{array}{*{20}{c}} 1&0&4 \\ 3&5&{ - 1} \\ 0&1&2 \end{array}} \right|\)
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Write minors and cofactors of the element of \(\left| {\begin{array}{*{20}{c}} 1&0&4 \\ 3&5&{ - 1} \\ 0&1&2 \end{array}} \right|\)

Answer

Let \(\Delta = \left| {\begin{array}{*{20}{c}} 1&0&4 \\ 3&5&{ - 1} \\ 0&1&2 \end{array}} \right|\) 
M11 = Minor of \({a_{11}} = \left| {\begin{array}{*{20}{c}} 5&{ - 1} \\ 1&2 \end{array}} \right| = 10 - \left( { - 1} \right) = 11\) and \({A_{11}} = {\left( { - 1} \right)^{1 + 1}}{M_{11}} = {\left( { - 1} \right)^2}\left( {11} \right) = 11\) 
M12 = Minor of \({a_{12}} = \left| {\begin{array}{*{20}{c}} 3&{ - 1} \\ 0&2 \end{array}} \right| = 6 - 0 = 6\) and \({A_{12}} = {\left( { - 1} \right)^{1 + 2}}{M_{12}} = {\left( { - 1} \right)^3}\left( 6 \right) = - 6\)
M13 = Minor of \({a_{13}} = \left| {\begin{array}{*{20}{c}} 3&5 \\ 0&1 \end{array}} \right| = 3 - 0 = 3\) and \({A_{13}} = {\left( { - 1} \right)^{1 + 2}}{M_{13}} = {\left( { - 1} \right)^4}\left( 3 \right) = 3\) 
M21 = Minor of \({a_{21}} = \left| {\begin{array}{*{20}{c}} 0&4 \\ 1&2 \end{array}} \right| = 0 - 4 = - 4\) and \({A_{21}} = {\left( { - 1} \right)^{2 + 1}}{M_{21}} = {\left( { - 1} \right)^3}\left( { - 4} \right) = 4\) 
M22 = Minor of \({a_{22}} = \left| {\begin{array}{*{20}{c}} 1&4 \\ 0&2 \end{array}} \right| = 2 - 0 = 2\) and \({A_{22}} = {\left( { - 1} \right)^{2 + 2}}{M_{22}} = \left( { - {1^4}} \right)\left( 2 \right) = 2\) 
M23 = Minor of \({a_{23}} = \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right| = 1 - 0 = 1\) and \({A_{23}} = {\left( { - 1} \right)^{2 + 2}}{M_{23}} = {\left( { - 1} \right)^5}\left( 1 \right) = - 1\) 
M31 = Minor of \({a_{31}} = \left| {\begin{array}{*{20}{c}} 0&4 \\ 5&{ - 1} \end{array}} \right| = 0 - 20 = - 20\) and \({A_{31}} = {\left( { - 1} \right)^{2 + 2}}{M_{31}} = {\left( { - 1} \right)^4}\left( { - 20} \right) = - 20\) 
M32 = Minor of \({a_{32}} = \left| {\begin{array}{*{20}{c}} 1&4 \\ 3&{ - 1} \end{array}} \right| = - 1 - 12 = - 13\) and \({A_{32}} = {\left( { - 1} \right)^{3 + 2}}{M_{32}} = {\left( { - 1} \right)^5}\left( { - 13} \right) = 13\) 
M33 = Minor of \({a_{33}} = \left| {\begin{array}{*{20}{c}} 1&0 \\ 3&5 \end{array}} \right| = 5 - 0 = 5\) and \({A_{33}} = {\left( { - 1} \right)^{3 + 3}}{M_{33}} = {\left( { - 1} \right)^6}\left( 5 \right) = 5\)
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