Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN: 9780534380588
Author: Wayne L. Winston
Publisher: Brooks Cole
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Chapter 2.1, Problem 7P

Explanation of Solution

a.

Trace of the given matrix:

The trace of a matrix is the sum of its diagonal elements.

Now, consider a matrix A of order n×n having elements aij and another matrix B of same order having elements bij.

Now, the addition of these matrices is,

A+B=[aij+bij]

The trace of this matrix is given below:

trace(A+B)=i=1n[aii+bii]

We can write this as follows:

  

Explanation of Solution

b.

Proof:

Consider any n×n matrices A and B.

Suppose C=AB.

Then the elements of this matrix are cij.

Then we can write,

cij=k=1n[aikbkj]

Similarly, let D=BA.

Then the elements of this matrix are dij.

Then we can write,

dij=k=1n[bikakj]

Now, the trace of the matrix AB is given below:

trace(AB)=i=1ncii

      =i=1n(k=1naikbki<

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