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Is the answer and explaination given correct ?

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A is correct.

 Here P is non singular of order 4 i.e $|P|  \neq 0$. So rank of P is 4

Assuming A and B are multiplication compatible. (which here they are)

so rank(AD) $\leq$ min(2,4)

rank(AD)  $\leq$ 2.

But the column rank of a matrix is invariant under pre multiplication by a non singular matrix. Similarly the row rank of a matrix is invariant under post multiplication by a non singular matrix.

So it will be 2.

Ref:  www.biostat.jhsph.edu › ~iruczins › teaching › cr.intro.pdf

 

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