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Question
If A is a 3 × 3 matrix, \[\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|\] then write the value of k.
Solution
\[\text{ Let }A = \begin{bmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{bmatrix} . \]
\[\text{ then, } 3A = \begin{bmatrix}3 a_1 & 3 a_2 & 3 a_3 \\ 3 b_1 & 3 b_2 & 3 b_3 \\ 3 c_1 & 3 c_2 & 3 c_3\end{bmatrix} . \]
\[\left| 3A \right| = \begin{vmatrix}3 a_1 & 3 a_2 & 3 a_3 \\ 3 b_1 & 3 b_2 & 3 b_3 \\ 3 c_1 & 3 c_2 & 3 c_3\end{vmatrix}\]
\[ = 3^3 \begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix} \left[\text{ Taking 3 common from ach row }\right]\]
\[ = 27\left| A \right|\]
Hence, the value of k is 27.
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