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Question:

If $A$ is a $3 \times 3$ matrix, $|A| \neq 0$ and $|3 A|=k|A|$ then write the value of $k$.

Solution:

Let $A=\left[\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right]$.

then, $3 A=\left[\begin{array}{lll}3 a_{1} & 3 a_{2} & 3 a_{3} \\ 3 b_{1} & 3 b_{2} & 3 b_{3} \\ 3 c_{1} & 3 c_{2} & 3 c_{3}\end{array}\right]$.

$|3 A|=\left|\begin{array}{lll}3 a_{1} & 3 a_{2} & 3 a_{3} \\ 3 b_{1} & 3 b_{2} & 3 b_{3} \\ 3 c_{1} & 3 c_{2} & 3 c_{3}\end{array}\right|$

$=3^{3}\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right|$                 [Taking 3 common from each row]

$=27|A|$

Hence, the value of $k$ is 27 .