Consider Δ = |2a 2b 2c; 2e f g; 2i j k|. The value of Δ = 4 × |a b c; e f g; i j k|

Assertion: Consider $\Delta = \begin{vmatrix} 2a & 2b & 2c \\ 2e & f & g \\ 2i & j & k \end{vmatrix}$. The value of $\Delta = 4 \times \begin{vmatrix} a & b & c \\ e & f & g \\ i & j & k \end{vmatrix}$

Reason: If all elements of one row or one column of a determinant are multiplied by a scalar, $k$ then the value of the determinant is multiplied by $k$. (Analysis) Which of the following is correct?

(a)Both Assertion and Reason are true, and Reason is the correct explanation for Assertion.
(b)Both Assertion and Reason are true, but Reason is not the correct explanation for Assertion.
(c)Assertion is true and Reason is false.
(d)Assertion is false and Reason is true.

Answer

Official answer key

Correct option: (d)

Answer: (d) Assertion is false, and Reason is true. The Reason is a standard property of determinants. In the Assertion only the first row and part of the first column are doubled (not a whole second or third row or column), so factoring gives $\Delta = 2\begin{vmatrix} a & b & c \\ 2e & f & g \\ 2i & j & k \end{vmatrix}$, which is not $4$ times the given determinant in general. The Assertion is false.