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Q.

 The trace Tr( A) of a 3×3 matrix A=aij is defined by the  relation Tr(A)=a11+a22+a33 (i.e., Tr(A) is sum of the main  diagonal elements). Which of the following statements  cannot hold '? 

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a

TrA2=Tr(A)2

b

TrI3=3

c

Tr(kA)=kTr( A)    (k is a scalar )

d

Tr(A+B)=Tr(A)+Tr(B)

answer is D.

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Detailed Solution

 (a) Tr(kA)=ka11+a22+a33=kTr(A)

 (b) Tr(A+B)=a11+b11+a22+b22+a33+b33=TrA+TrB

cTrI3=1+1+1=3

dTrA2=Σa112+Σa122a11+a22+a332=TrA2

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 The trace Tr( A) of a 3×3 matrix A=aij is defined by the  relation Tr(A)=a11+a22+a33 (i.e., Tr(A) is sum of the main  diagonal elements). Which of the following statements  cannot hold '?