Question

Let $△=\left|\begin{array}{ccc}-f\left(x\right)& g\left(x\right)& -h\left(x\right)\\ g\left(y\right)& -h\left(y\right)& f\left(y\right)\\ -h\left(z\right)& f\left(z\right)& -g\left(z\right)\end{array}\right|,$ 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. $g(-x)M_{12}+h(-y)M_{22}+f(-z)M_{32}=△$
• B. $g\left(x\right)C_{12}+h\left(y\right)C_{22}+f\left(z\right)C_{32}=△$
• C. $f(-x)M_{31}+g(-x)M_{32}+h(-x)M_{33}=0$
• D. $f\left(x\right)C_{31}+g(-x)C_{32}+h\left(x\right)C_{33}=0$

Answer 1

The correct option is B $g\left(x\right)C_{12}+h\left(y\right)C_{22}+f\left(z\right)C_{32}=△$

we know, $△=$ sum of product of elements in any row/column to its corresponding co-factors.
Expanding along $C_2:-g\left(x\right)M_{12}-h\left(y\right)M_{22}-f\left(z\right)M_{32}=△$
$\Rightarrow g(-x)M_{12}+h(-y)M_{22}+f(-z)M_{32}=△$
or
$\Rightarrow g\left(x\right)C_{12}+h(-y)C_{22}+f\left(z\right)C_{32}=△$
Also, sum of product of elements of any row/column to corresponding co-factors of other row/column is zero.
Expanding along $R_1:$
$f(-x)M_{31}+g(-x)M_{32}+h(-x)M_{33}=0$
This can be written as
$\Rightarrow -f\left(x\right)C_{31}+g\left(x\right)C_{32}-h\left(x\right)C_{33}=0\Rightarrow f\left(x\right)C_{31}+g(-x)C_{32}+h\left(x\right)C_{33}=0$