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Chapter 3: Q5P (page 141)

Show that the product AAΤis a symmetric matrix.

Short Answer

Expert verified

It is shown that AAΤis a symmetric matrix.

Step by step solution

01

Given information

The given product is AAΤ, where Aand AΤ are two matrices.

02

Definition of a symmetric matrix

A matrix Ais said to be symmetric if A=AΤ, where T represents transpose of a matrix.

03

To show that   is a symmetric matrix

To prove the product AAΤis a symmetric matrix, it is necessary to show that (AAT)=AATT.

It is known that, ABT=BTAT. Therefore, AAT=AATTcan be written as follows:

AATT=ATTAT=AAT∵ATT=A

Thus, AAT=AATT

Hence, the product AAT is a symmetric matrix.

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Most popular questions from this chapter

Question: For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using

6.{x+y-z=13x+2y-2z=3

In Problems 8 to 15,use to show that the given functions are linearly independent.

15.ex,eix,cosh⁡x

For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using.

3.

solve the set of equations by the method of finding the inverse of the coefficient matrix.

{2x+3y=-15x+4y=8

Find the symmetric equations (5.6) or (5.7) and the parametric equations (5.8) of a line, and/or the equation (5.10) of the plane satisfying the following given conditions.

Line through and parallel to the line .

Answer

The symmetric equations of the line is .

The parametric equation is .

Step-by-Step Solution

Step 1: Concept of the symmetric and parametric equations

The symmetric equations of the line passing through and parallel to is

The parametric equations of the line are

Step 2: Determine the symmetric equation of a straight line

The given point is and the line is .

The given line is in the form of . So, we get

The symmetric equations of the straight line passing through and parallel to is given by

Thus, the required solution is .

Step 3: Determine the parametric equation of a straight line.

The parametric equations of the straight line passing through and parallel to is given by

Or

.

Thus, the required solution is .
See all solutions

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