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In a matrix $A$, if all the sub matrices or $k$ orcer are singular and there is one non-singular sub matrix of order r $(r < k)$, then the rank $(\rho)$ of the matrix A
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satisfies $r \leq \rho < k$
Since, all the sub-matrices of $k^{\text {th }}$ order are singular.
$\therefore \quad$ Rank of $A=\rho < k$
$\Rightarrow \rho < k$ ...(i)
Also, there is one non-singular sub matrix of order $r$.
So, $r \leq$ Rank of $A$
$\Rightarrow r \leq \rho$ ...(ii)
From eqn. (1) and (11), we get
$r \leq \rho < k$
$\therefore \quad$ Rank of $A=\rho < k$
$\Rightarrow \rho < k$ ...(i)
Also, there is one non-singular sub matrix of order $r$.
So, $r \leq$ Rank of $A$
$\Rightarrow r \leq \rho$ ...(ii)
From eqn. (1) and (11), we get
$r \leq \rho < k$
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