Introduction to mathematical programming
Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
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Chapter 2.1, Problem 5P

Explanation of Solution

a.

Symmetric matrix:

An n×n matrix A is symmetric if A=AT.

Now, consider a matrix A of order n×n having elements aij, then the elements of this transpose matrix  AT will have elements aji.

Now the product AAT is defined because both the matrices are square matrices of same order and the product will also be of the same order square matrix.

If the elements of this product matrix are bij, then from the definition of the product of matrices we have bij= Scalar product of row i of A and column j of AT. But we know that row i of A is same as the column j of AT.

To show this, consider a 2×2 matrix as given below:

A=[a11a12a21a22]

Then transpose is as follows:

AT=[a11a21a12a22]

Therefore, multiplication is,

AAT=[a11a12a21a22

Explanation of Solution

b.

Symmetric matrix:

Now taking the same matrix as in part (a.), we have

A+AT=[a11a12a21a22]+[a11a21a12a22]

  =[2a11a12+a21

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