Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants2 Marks
Question
Write the value of $a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23}.$
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Answer
We know that in a square matrix of order $n,$ the sum of the products of elements of a row $($or a column$)$ with the cofactors of the corresponding elements of some other row $($or column$)$ is zero. Therefore,
$A = [a_{ij}] $ is a square matrix of order $n.$
$\Rightarrow\sum\limits_{\text{n}}^{\text{i}=1}\text{a}_{\text{ij}}\text{C}_\text{kj}=0$ and $\sum\limits_{\text{i}=1}^{\text{n}}\text{a}_{\text{ij}}\text{C}_\text{ik}=0$
$\Rightarrow a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23} = 0$
$[$Since the elements are of first row and the cofactors are of elements of second row$]$
$\Rightarrow a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23} = 0$
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