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Q.

 Let P1=I=100010001,P2=100001010,P3=010100001,P4=010001100,P5=001100010,P6=001010100 and X=k=16Pk213102321PkTWhere PkT denotes the transpose of the matrix Pk. Then which of the following options is/ are correct?

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a

  The sum of diagonal entries of X is 18

b

X is a symmetric matrix

c

If X111=α111, then α=30

d

 X – 30I is an invertible matrix

answer is A, B, D.

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Detailed Solution

 Let A=213102321X=k=16PkAPkT, Here PkPkT=PkTPk=I for all k=1,2,3,4,5,6 . PkT=Pk-1&AT=A and    XT=k=16PkAPkTT=k=16PkAPkT=X

 Let B=111XB=K=16PKAPKTB=K=16PKAB=K=16PKAB=222222222AB=303030=30111=30Bα=30 and (X-30I) is singular,   Trk=16PkAPkT=18

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