Question

$\text{If}\left|\begin{array}{ccc}ka& kb& kc\\ kd& ke& kf\\ kg& kh& ki\end{array}\right|=A\text{and}$
$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=B$
Then the value of $A/B$ is

• A. 1
• B. $k$
• C. $k^3$
• D. 0

Answer 1

The correct option is C

$k^3$

It’s the property of the determinant that a scalar multiplied to a determinant will be same as the same scalar being multiplied to elements of one of the row or column of the same determinant. So if the scalar is multiplied to 3 rows or columns you can take out the scalar 3 times.
Given
$\left|\begin{array}{ccc}ka& kb& kc\\ kd& ke& kf\\ kg& kh& ki\end{array}\right|=A$

$\Rightarrow A=k^3$$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=k^{3}B$

$\therefore A/B=k^3$