Question

If A is a 3 × 3 matrix, $\left|A\right|\ne 0\mathrm{and}\left|3A\right|=k\left|A\right|$ then write the value of k.

Answer 1

$$\begin{gathered} \mathrm{Let}A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]. \\ \mathrm{then},3A=\left[\begin{array}{ccc}3a_1& 3a_2& 3a_3\\ 3b_1& 3b_2& 3b_3\\ 3c_1& 3c_2& 3c_3\end{array}\right]. \\ \left|3A\right|=\left|\begin{array}{ccc}3a_1& 3a_2& 3a_3\\ 3b_1& 3b_2& 3b_3\\ 3c_1& 3c_2& 3c_3\end{array}\right| \\ =3^3\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|\left[\mathrm{Taking}3\mathrm{common}\mathrm{from}\mathrm{each}\mathrm{row}\right] \\ =27\left|A\right| \\ \mathrm{Hence},\mathrm{the}\mathrm{value}\mathrm{of}k\mathrm{is}27. \end{gathered}$$