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Which one of the following equations is a correct identity for arbitrary $3  \times 3$ real matrices $P$, $Q$ and $R$?

  1. $P(Q+R)=PQ+RP$
  2. $(P-Q)^2 = P^2 -2PQ -Q^2$
  3. $\text{det } (P+Q)= \text{det } P+  \text{det } Q$
  4. $(P+Q)^2=P^2+PQ+QP+Q^2$
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Ans -d

Option a)  P (Q+R)=PQ+QR is wrong because matrix  multiplication  is not commutative .

option b) ( P-Q)^2 should be P^2 -PQ-QP +Q^2

 WE can't write 2pq here because PQ and QP are not same in matrix  multiplication.

Option c) it is not the property  of determinant .

Option d has written  correctly  here.so answer  is option  d
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