If A= [ $a_{i j}$ ] is a matrix of $2 \times 2$ order such that $| A | = - 15$ and $C_{i j}$ represents the co-factors of $a_{i j}$ then find the value of $a_{21} C_{21} + a_{22} C_{22}$

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Solution

Given the matrix
$A = [\begin{matrix} a_{11} & a_{12} a_{21} & a_{22} \end{matrix}]$ .
Now, $| A | = a_{11} a_{22} - a_{21} . a_{12}$ .......(1).
Given $C_{i j}$ is the co-factor of $a_{i j}$ .
Now, $C_{21} = - a_{12}$ and $C_{22} = a_{11}$ .
$∴ a_{21} C_{21} + a_{22} C_{22}$
$= - a_{21} a_{12} + a_{22} a_{11}$
$= | A |$ [using (1)]
$= - 15$

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