Question

Let $△=\left|\begin{array}{ccc}f\left(x\right)& -g\left(x\right)& h\left(x\right)\\ -x& y& z\\ -f\left(y\right)& g\left(y\right)& -h\left(y\right)\end{array}\right|,$ then which of the following statement is not correct?
(where $M_{ij}$ and $C_{ij}$ are minor and co-factor of element $a_{ij}$ and $f,g,h$ are odd functions)

• A. $x{C}_{11}-y{C}_{12}-z{C}_{13}=0$
• B. $f\left(x\right)C_{31}+g(-x)C_{32}+h\left(x\right)C_{33}=0$
• C. $f(-y)M_{11}+g\left(y\right)M_{12}+h(-y)M_{13}=0$
• D. $f(-y)C_{11}+g\left(y\right)C_{12}+h(-y)C_{13}=0$

Answer 1

The correct option is C $f(-y)M_{11}+g\left(y\right)M_{12}+h(-y)M_{13}=0$

Given : $△=\left|\begin{array}{ccc}f\left(x\right)& -g\left(x\right)& h\left(x\right)\\ -x& y& z\\ -f\left(y\right)& g\left(y\right)& -h\left(y\right)\end{array}\right|$
We know, sum of product of elements of any row/column to corresponding co-factors of other row/colunm is zero.
$\Rightarrow f\left(x\right)C_{31}-g\left(x\right)C_{32}+h\left(x\right)C_{33}=0\Rightarrow f\left(x\right)C_{31}+g(-x)C_{32}+h\left(x\right)C_{33}=0$

and $-f\left(y\right)C_{11}+g\left(y\right)C_{12}-h\left(y\right)C_{13}=0\Rightarrow f(-y)C_{11}+g\left(y\right)C_{12}+h(-y)C_{13}=0$
or,$f(-y)M_{11}-g\left(y\right)M_{12}+h(-y)M_{13}=0$

and $-x{C}_{11}+y{C}_{12}+z{C}_{13}=0\Rightarrow x{C}_{11}-y{C}_{12}-z{C}_{13}=0$