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Question

For any 3×3 matrix M, let |M| denote the determinant of M. Let E=12323481318, P=100001010 and F=13281813243. If Q is a nonsingular matrix of order 3×3, then which of the following statements is(are) TRUE?

A
F=PEP and P2=100010001
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B
|EQ+PFQ1|=|EQ|+|PFQ1|
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C
|(EF)3|>|EF|2
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D
Sum of the diagonal entries of P1EP+F is equal to the sum of diaginal entries of E+P1FP
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Solution

The correct option is D Sum of the diagonal entries of P1EP+F is equal to the sum of diaginal entries of E+P1FP
Clearly, P2=100010001=1 and
PE=12381318234
PEP=13281813243=F


|P||E||P|=|F|
|F|=0 and
|E|=∣ ∣12323481318∣ ∣
R3R33R22R1
|E|=∣ ∣123234000∣ ∣
|E|=0
Now, |EQ+PFQ1|
=|EQ+PPEPQ1|
=|E||Q+PQ1|
=0 (|E|=0)
Also, |EQ|+|PFQ1|
|E||Q|+|P||F||Q1|
=0 (|E|=0, |F|=0)
|EQ+PFQ1|=|EQ|+|PFQ1|


|(EF)3|=|E|3|F|3=0
|EF|2=|E|2|F|2=0
|(EF)3|=|EF|2


P1EP+F
PEP+F (P2=I)
F+F=2F
Tr(2F)=2×22=44
and
E+P1FP
E+P1PEPP (F=PEP)
E+E=2E
Tr(2E)=2×22=44

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