Question

If A and B are non-singular square matrices of order n such that A = kB, then $\frac{\left|A\right|}{\left|B\right|}=$ ____________.

Answer 1

Given:
A and B are non-singular square matrices of order n.
A = kB


$$\begin{gathered} A=kB \\ \mathrm{Taking}\mathrm{determinant}\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \left|A\right|=\left|kB\right| \\ \Rightarrow \left|A\right|=k^n\left|B\right|\left(\because \mathrm{Order}\mathrm{of}B\mathrm{is}n\times n\right) \\ \Rightarrow \frac{\left|A\right|}{\left|B\right|}=k^n \end{gathered}$$

Hence, $\frac{\left|A\right|}{\left|B\right|}=$kⁿ.