AP EAMCET · Maths · Matrices
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 \(\mathrm{P}\)
- A satisfies \(\mathrm{k} \leq \rho \leq \mathrm{m}\)
- B satisfies \(\mathrm{k} < \rho < \) n
- C satisfies \(\mathrm{k} \leq \rho \leq \min \{\mathrm{m}, \mathrm{n}\}\)
- D is equal to \(\mathrm{k}+1\)
Answer & Solution
Correct Answer
(C) satisfies \(\mathrm{k} \leq \rho \leq \min \{\mathrm{m}, \mathrm{n}\}\)
Step-by-step Solution
Detailed explanation
\(\because\) 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 :…
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