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Chapter 7: Q42E (page 338)

Consider twomatrices A and B such that BA = 0show thattr((A+B)2)=tr(A2)+tr(B2)Hint: exercise is helpful.

Short Answer

Expert verified

Consider two matrices.

tr((A+B)2)=tr(A2)+tr(B2)=tr(A2)+tr(B2)=LHS=RHS

Step by step solution

01

definition of matrix

A function is defined as a relationship between a set of inputs that each have one output.

Given,

tr((A+B)2)=tr(A2)+tr(B2)LHS=tr((A+B)2)=tr((A+B)(A+B))=tr(A2+BA+AB+B2)=tr(A2)+tr(BA)+tr(AB)+tr(B2)

02

definition of A and B such that BA = 0 show that

BA is a zero matrix.

BA = 0

As a zero matrix = 0

tr (BA) = 0

Used the result is a 40

tr (BA) = tr (AB)

A and B Are n脳n

tr(AB)=tr(BA)=0tr((A+B)2)=tr(A2)+0+0+tr(B2)=tr(A2)+tr(B2)

Hence,

tr((A+B)2)=tr(A2)+tr(B2)=tr(A2)+tr(B2)=LHS=RHS

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