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I was thinking about the above problem.Can someone point me in the right direction? It appears that option $(c)$ is correct.But i am looking for suitable example to establish it.Thanks everyone in advance for your time.

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up vote 3 down vote accepted

Notice that $PQR$ is a $3 \times 3$-matrix equal to the identity, this means that it has rank $3$. But $Q$ can at most have rank $2$, so $PQR$ has rank at most two, contradicting it being the identity.

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I think if the rank of $P$ be $2$ so there is a zero row in $P$, for example $$R_3(P):=[0,0]$$ If we accept this so as we go to find $PQR$ the third row of the final matrix woul be full of zero. It contradicts our assumption. So I think a and c cannot be true.

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Nice and short! +1 – amWhy Feb 17 '13 at 0:03

For any matrix product $AB$ (if it exists), we have $\mathrm{rank}(AB) \leq \mathrm{min}\{\mathrm{rank}(A), \mathrm{rank}(B)\}$. Can you see how this leads to a correct answer to the question?

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