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Determinants — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDeterminants2 Marks
Question
Evaluate the following determinant:
$\begin{vmatrix}67&19&21\\39&13&14\\81&24&26 \end{vmatrix}$
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Answer

Consider the determinant
$\triangle=\begin{vmatrix}67&19&21\\39&13&14\\81&24&26 \end{vmatrix}$
Applying $C_1 \rightarrow C_1- 4C_3$​​​​​​​, We get,
$\triangle=\begin{vmatrix}4&19&21\\-3&13&14\\-3&24&26 \end{vmatrix}$
$\Rightarrow\triangle=\begin{vmatrix}4&19&21\\-3&13&14\\-3&24&26 \end{vmatrix}$
$\Rightarrow\triangle=\begin{vmatrix}1&32&35\\-3&13&14\\0&11&12\end{vmatrix}$ [Applying $R_3 \rightarrow R_3 - R_2​​​​​​​$​​​​​​​ and $R_1 \rightarrow R_1 + R_2$​​​​​​​]
$\Rightarrow\triangle=\begin{vmatrix}1&32&35\\0&109&119\\0&11&12\end{vmatrix}$ [Applying $R_2 \rightarrow 3R_1 + R_2​​​​​​​$​​​​​​​]
$\Rightarrow\triangle=1(109\times12-119\times11)$
$\Rightarrow\triangle=-1$

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