Question

Assertion :The determinant of $A=\left[\begin{array}{ccc}1& 2& 3\\ 5& 6& 7\\ 2& 4& 6\end{array}\right]$is zero Reason: The determinant of a matrix vanishes if any two rows or columns are proportional.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R} is false,
• D. (A)is false but (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

$A=\left[\begin{array}{ccc}1& 2& 3\\ 5& 6& 7\\ 2& 4& 6\end{array}\right]$
$\therefore \left|A\right|=\left|\begin{array}{ccc}1& 2& 3\\ 5& 6& 7\\ 2& 4& 6\end{array}\right|$
$C_1\to C_1+C_3-C_2$
$=\left|\begin{array}{ccc}2& 2& 3\\ 6& 6& 7\\ 4& 4& 6\end{array}\right|$
$\therefore |A|=0(\because C_1$ and $C_2$ are identical)
$\therefore$ Assertion (A) and Reason (R) are individually true and Reason (R) is correct explanation of Assertion (A).