Question

Let $P$ and $Q$ be two matrices different from identity matrix $I$ and null matrix $O$ such that $PQ=QP$ and if $P^6-Q^6=P^5-Q^5=P^4-Q^4=I$ then,

• A. $I-P$ is singular
• B. $I-Q$ is singular
• C. $P+Q=PQ$
• D. $(I-P)(I-Q)$ is non singular

Answer 1

Answer: (A), (B)

$I-Q$ is singular

$P^6-Q^6=(P+Q)(P^5-Q^5)-PQ(P^4-Q^4)$
$I=(P+Q)I-\left(PQ\right)$
$P+Q=PQ+I\Rightarrow P+Q\ne PQ$
$\Rightarrow I+PQ-(P+Q)=0$
$\Rightarrow I^2-(P+Q)I+PQ=0$
$\Rightarrow (I-P)(I-Q)=0$
But $P\ne I,Q\ne I$
$\Rightarrow$ Product of $(I-P),(I-Q)$ is $0$

Suppose, we have $A$ and $B$ matrices such that -
$AB=0,A\ne 0,B\ne 0$
Assume, $|A|\ne 0,B=IB=A^{-1}AB=0$ but $B\ne 0$ (Therefore, our assumption is wrong)
$\Rightarrow |A|=0\text{and}|B|=0$
$\Rightarrow (I-P)(I-Q)=0$
$\Rightarrow (I-P)\text{\&}(I-Q)$ are singular matrices.