# Find an appropriate matrix according to specific conditions

• Avibu
In summary, the conversation discusses finding matrices that satisfy certain conditions, including being eigenvectors with specific eigenvalues. The attempt at a solution involves using equations and matrices, but the conversation ends with the realization that there are no appropriate matrices that match the given conditions. The individual expresses gratitude for the assistance and clarification.
Avibu
I am facing some difficulties solving one of the questions we had in our previous exam. I am sorry for the bad translation , I hope this is clear.

In each section, find all approppriate matrices 2x2 (if exists) , which implementing the given conditions:

• is an eigenvector of A with eigenvalue of 10 , and
is an eigenvector of A with eigenvalue of 20
.

• is an eigenvector of A with eigenvalue of 10 , and EXISTS eigenvector of A with eigenvalue of 20
If there are no matrices matched , explain why.

## The Attempt at a Solution

[/B]
I tried to build equations for the first section but I have no idea how to keep from there :

Thanks.

What is the relation between ##(1, 3)^T## and ##(2, 6)^T##?

I am not sure if I understood your question but the vectors seem to be linearly dependent

Avibu said:
I am not sure if I understood your question but the vectors seem to be linearly dependent
Exactly. So what happens when you apply ##A## to those vectors?

Avibu
If I put those equations from step 3 (above) in a matrix , I will have 2 rows filled with 0's and Rank A < Rank(A|b) => No solution?
I would apprciate if you could explain it better than I do ,becasue I really want to understand what I am doing and how it should be solved.

Keep it simpler. You have a vector ##v## such that
$$A v = 10 v$$
Take a second vector ##u = 2 v##. What is ##Au##?

Ok so, u=2v

{ Av = 10v
{ Au = 20u

{ Av = 10v
{ A(2v) = 20(2v)

{ Av = 10v
{ 2(Av) = 40v

{ Av = 10v
{ Av = 20v

I think I missed the point , I can see what you are trying to do but still can't figure it out

Avibu said:
Ok so, u=2v
{ Av = 10v
{ Av = 20v

Avibu
It is ! :)
It means there are no appropriate matrices.

## What is a matrix and why is it important?

A matrix is a rectangular array of numbers or symbols that is commonly used in mathematics, science, and engineering to represent data or relationships between variables. It is important because it allows for the efficient organization and manipulation of complex information.

## What are the different types of matrices?

There are several types of matrices, including square matrices (having an equal number of rows and columns), diagonal matrices (containing non-zero elements only on the main diagonal), and symmetric matrices (whose elements are equal to their corresponding elements across the main diagonal).

## What are the conditions that determine an appropriate matrix?

The conditions that determine an appropriate matrix depend on the specific problem or application. Some common conditions include the desired dimensions of the matrix, the type of data being represented, and any mathematical operations or transformations that need to be performed on the matrix.

## How do I find an appropriate matrix that meets certain conditions?

To find an appropriate matrix, you first need to identify the specific conditions that it must meet. Then, you can use mathematical techniques such as matrix operations, transformations, or algorithms to create a matrix that satisfies those conditions.

## What are some applications of using matrices?

Matrices have a wide range of applications, including data analysis, computer graphics, physics, and economics. They are also used in solving systems of equations, representing transformations in geometry, and performing calculations in quantum mechanics.

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