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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 $\mathrm{P}$
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satisfies $\mathrm{k} \leq \rho \leq \min \{\mathrm{m}, \mathrm{n}\}$
$\because$ 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 :
$k \leq \rho \leq \min (m, n)$
$\therefore$ 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 :
$k \leq \rho \leq \min (m, n)$
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