Question

2. If A ig a non-zero 3 x 3 matrix such that |5A|= k|A|find k​

Answer 1

Given that,

A is a non-zero 3 x 3 matrix such that |5A|= k|A|.

As we know that,

if A is a matrix of order n × n then |kA| = kⁿ|A|.

In the given conditions,

|5A| = 5ⁿ|A|

⇒ |5A| = 5³|A|                         [∵ n = 3]

⇒ |5A| = 125|A|

Hence, k = 125.

______________________________

If A is a square matrix |kA| of order n = 3,

$\sf{|A|=}\left[\begin{array}{c c c} \sf{a} & \sf{b} & \sf{c} \\ \sf{d} & \sf{e} & \sf{f} \\ \sf{i} & \sf{j} & \sf{k} \end{array}\right]$

$\longrightarrow \sf{|kA|=}\left[\begin{array}{c c c} \sf{ka} & \sf{kb} & \sf{kc} \\ \sf{kd} & \sf{ke} & \sf{kf} \\ \sf{ki} & \sf{kj} & \sf{kk} \end{array}\right]$

$\longrightarrow \sf{|kA|=k^3}\left[\begin{array}{c c c} \sf{a} & \sf{b} & \sf{c} \\ \sf{d} & \sf{e} & \sf{f} \\ \sf{i} & \sf{j} & \sf{k} \end{array}\right]$

$\longrightarrow \sf{|kA|=k^3|A|}$

$\boxed{\longrightarrow \sf{|kA|=k^n|A|}}$