Find a 2x2 Matrix which performs the operation....

• Jtechguy21
In summary, the problem asks for a matrix that when multiplied by e1 will give e2, and when multiplied by e2 will give e1. The problem asks for a matrix that when multiplied by e1 will give e2, and when multiplied by e2 will give e1.
Jtechguy21

Homework Statement

[/B]Find the matrix that performs the operation

2x2 Matrix which sends e1→e2 and e2→e1

The Attempt at a Solution

[/B]
I know e1 = < 1 , 0>
and e2 = <0 , 1>

Basically I'm not quite sure what the question is asking. This is the one of the problems I am currently stuck on. He is teaching out of the book-currently

Can someone please explain what this means?

The problem asks you to find a matrix ##A## such that ##A e_1 = e_2## and ##Ae_2 = e_1##.

Orodruin said:
The problem asks you to find a matrix ##A## such that ##A e_1 = e_2## and ##Ae_2 = e_1##.

thanks. So If i understand from your clarification,

This 2x2 matrix that I'm looking for, when I multiply it by e1 it will give me e2, and vice-versa, this unknown matrix multipled by e2 will give me the result e1.

Yes.

Orodruin said:
Yes.
Thanks again.

When i find the 2 these two matrixes that perform these two operations. When I multiply them together the result should be the 2x2 matrix that this question is asking for?

No. You need to find a single matrix which satisfies both relations.

Orodruin said:
No. You need to find a single matrix which satisfies both relations.

How is that even possible?
1] I found a 2x2 matrix we'll call A that when you multiply it by e1 = e2
2] I also found a 2x2 matrix B that when you multiply it by e2 =e1

But how is it possible to satisfy both conditions at the same time with the same 2x2 matrix? Any hints?

Any matrix is defined by its action on a complete basis. If you just know how it acts on a single vector you cannot determine it uniquely. I suggest assuming the most general form
$$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$
and examine what you can say about ##a,b,c## and ##d## from your requirements.

Orodruin said:
Any matrix is defined by its action on a complete basis. If you just know how it acts on a single vector you cannot determine it uniquely. I suggest assuming the most general form
$$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$
and examine what you can say about ##a,b,c## and ##d## from your requirements.

Nevermind I just figured it out. thanks for your help with adding to my knowledge/intuition with that statement about defining a matrix actions.
!(took me long enough)

The 2x2 matrix i came up with was the following.

Like Orodruin was saying, a matrix associated to a linear map T is uniquely defined by the effects of T on the given choice of basis. Then , for a map (Assummiing this, given your posts) from ##\mathbb R^2 ## to itself , both with the same basis, the associated matrix is ## [(Te_1)^T , (Te_2)^T] ##. where the T means transpose. Note that the copies of ##\mathbb R^2 ## may have different bases, then the solution is different. There are many variants, of course, of maps between vector spaces ( or even --free, of course-- rings/modules) of different dimensions, with different bases, but the idea of your case generalizes nicely here.

Last edited:

1. What is a 2x2 matrix?

A 2x2 matrix is a rectangular array of numbers arranged in 2 rows and 2 columns. It is commonly used in mathematics, physics, and other scientific fields to represent and manipulate data.

2. How do I perform an operation using a 2x2 matrix?

To perform an operation using a 2x2 matrix, you need to follow certain rules depending on the type of operation. For example, to add or subtract two matrices, you need to add or subtract the corresponding elements in each matrix. To multiply two matrices, you need to multiply the elements in each row of the first matrix by the corresponding elements in each column of the second matrix, and then add the products together.

3. What types of operations can be performed using a 2x2 matrix?

A 2x2 matrix can be used to perform various operations such as addition, subtraction, multiplication, and division. It can also be used to find determinants, inverse matrices, and eigenvalues/eigenvectors.

4. Can a 2x2 matrix perform any operation?

No, a 2x2 matrix can only perform operations that are defined for matrices. For example, you cannot take the square root of a matrix or find its logarithm.

5. Why is it important to find a 2x2 matrix that performs a specific operation?

Finding a 2x2 matrix that performs a specific operation allows you to manipulate and understand data in a more efficient way. It also helps in solving mathematical equations and problems, and in creating models and simulations in various scientific fields.

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