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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants1 Mark
Question
Find adjoint of the matrix $\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|$
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Answer

Here $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]$$\Rightarrow \left| A \right| = \left| {\begin{array}{*{20}{c}} 1&2 \\ 3&4 \end{array}} \right|$
$\therefore A_{11} =$ Cofactor of ${a_{11}} = {\left( { - 1} \right)^2}\left( 4 \right) = 4$
$A_{12} =$ Cofactor of ${a_{12}} = {\left( { - 1} \right)^3}\left( 3 \right) = - 3$
$A_{21} =$ Cofactor of ${a_{21}} = {\left( { - 1} \right)^3}\left( 2 \right) = - 2$
$A_{22} =$ Cofactor of ${a_{22}} = {\left( { - 1} \right)^4}\left( 1 \right) = 1$ 
$\therefore adj_A = \left| {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}} \\ {{A_{21}}}&{{A_{22}}} \end{array}} \right|$
$= \left| {\begin{array}{*{20}{c}} 4&{ - 3} \\ { - 2}&1 \end{array}} \right|$

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