If there exists a $\mathrm{k}^{\text {th }}$ order non-singular sub matrix in matrix $\mathrm{P}$ of order…

If there exists a $\mathrm{k}^{\text {th }}$ order non-singular sub matrix in matrix $\mathrm{P}$ of order $\mathrm{m} \times \mathrm{n}$, then the rank $(\rho)$ of $\mathrm{P}$
  1. satisfies $\mathrm{k} \leq \rho \leq \mathrm{m}$
  2. satisfies $\mathrm{k} < \rho < $ n
  3. satisfies $\mathrm{k} \leq \rho \leq \min \{\mathrm{m}, \mathrm{n}\}$
  4. is equal to $\mathrm{k}+1$

Solution

$\because$ The order of the matrix $P$ is $m \times n$. $\therefore$ Rank of $P$ i.e. $\rho \leq \min (m, n)$ ...(i) Also, there exist a $k^{\text {th }}$ order non-singular sub matrix. $\Rightarrow \quad \rho \geq k$ ...(ii) Combining eqn. (i) and (ii), we get : $k \leq \rho \leq \min (m, n)$

Asked in: AP EAMCET 2023 (16 May Shift 1)

Practice more Matrices questions on Aicharya