Write the value of a11C21 + a12C22 + a13C23. - Vidyadip

Question

Write the value of a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C_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 $\sum\limits_{\text{i}=1}^{\text{n}}\text{a}_{\text{ij}}\text{C}_\text{ik}=0$
⇒ a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃ = 0
[Since the elements are of first row and the cofactors are of elements of second row]
⇒ a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃ = 0

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