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Cofactor

Let =f -g -...

Question

Let $△ = ∣ ∣ ∣ ∣ \begin{matrix} f & - g & - h - a & b & - c - x & y & - z \end{matrix} ∣ ∣ ∣ ∣,$ then which of the following statement(s) is(are) correct?
(where $M_{i j}$ and $C_{i j}$ are minor and co-factor of element $a_{i j}$ )

A

a M_{21} + b M_{22} + c M_{23} = △

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B

a C_{21} + b C_{22} + c C_{23} = △

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C

- h C_{13} + c C_{23} - z C_{33} = △

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D

- h M_{13} + c M_{23} - z M_{33} = △

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Solution

The correct option is D $- h M_{13} + c M_{23} - z M_{33} = △$
Given : $△ = ∣ ∣ ∣ ∣ \begin{matrix} f & - g & - h - a & b & - c - x & y & - z \end{matrix} ∣ ∣ ∣ ∣$
we know, $△ =$ sum of product of elements in any row/column to its corresponding co-factors.
Now, expanding along $R_{2} : - (- a) M_{21} + b M_{22} - (- c) M_{23} = △$
$\Rightarrow a M_{21} + b M_{22} + c M_{23} = △$
or $- a C_{21} + b C_{22} - c C_{23} = △$ and
Expanding along $C_{3} : - h M_{13} - (- c) M_{23} - z M_{33} = △$
or $- h C_{13} - c C_{23} - z C_{33} = △$

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Similar questions

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