Question

Choose the correct answer in questions

Let A be a square matrix of order $3\times 3$, then $|kA|$ is equal
a) $k|A|$
b) $k^2|A|$
c) $k^3|A|$
d) $3k|A|$

Answer 1

A is a square matrix of order $3\times 3$
Let $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right],Then,kA=\left[\begin{array}{ccc}k{a}_1& k{b}_1& k{c}_1\\ k{a}_2& k{b}_2& k{c}_2\\ k{a}_3& k{b}_3& k{c}_3\end{array}\right]$

$\therefore |kA|=\left|\begin{array}{ccc}k{a}_1& k{b}_1& k{c}_1\\ k{a}_2& k{b}_2& k{c}_2\\ k{a}_3& k{b}_3& k{c}_3\end{array}\right|$

Taking out common factor k from each row, we get
$=k^3\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|\therefore |kA|=k^3|A|.$
Hence, the correct answer is (c).