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Find adjoint of the matrix \(\left[ {\begin{array}{*{20}{c}} 1&2 \\ 3&4 \end{array}} \right]\)
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Find adjoint of the matrix \(\left[ {\begin{array}{*{20}{c}} 1&2 \\ 3&4 \end{array}} \right]\)

Answer

\(\begin{bmatrix} 4&-2\\-3&1\end{bmatrix}\)
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Solution

Let \(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}} \\ {{a_{21}}}&{{a_{22}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&2 \\ 3&4 \end{array}} \right]\)
Adjoint of a matrix is transpose of cofactor matrix
\(\therefore\) A11 = Cofactor of \({a_{11}} = {\left( { - 1} \right)^2}\left( 4 \right) = 4\)
A12 = Cofactor of \({a_{12}} = {\left( { - 1} \right)^3}\left( 3 \right) = - 3\)
A21 = Cofactor of \({a_{21}} = {\left( { - 1} \right)^3}\left( 2 \right) = - 2\)
A22 = Cofactor of \({a_{22}} = {\left( { - 1} \right)^4}\left( 1 \right) = 1\)
Cofactor matrix \(\begin{bmatrix} 4&-3\\-2&1\end{bmatrix}\)
So, \(adj A=\begin{bmatrix} 4&-2\\-3&1\end{bmatrix}\)
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