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Chapter 7: Q40E (page 338)

If A and B are n脳nmatrices, show that tr (AB) = tr (BA)

Short Answer

Expert verified

If A and B aren脳nmatrices.

AB=cijn脳nA=aijn脳nAndB=bijn脳n

From (1)

= tr (AB)

鈭�(BA)=tr(AB)

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,

AB=cijn脳nA=aijn脳nandB=bijn脳n

Then,

Multiplication of matrice:

Ascik=鈭�j-1naijbjk

鈭�tr(AB)=鈭�i-1nCii

02

definition of tr (BA) = tr (AB) 

BA and AB Equal the values are.

Similarly tr (BA)

tr(BA)=鈭�i-1n鈭�j-1nbijaji

As I and j are independent we can interchange them.

tr(BA)=鈭�i-1n鈭�j-1nbijaji=鈭�i-1n鈭�j-1nbijaji

from (1)

= tr (AB)

Hence,

鈭�(BA)=tr(AB)

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For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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