Proving Linear Algebra: AB=BA for All B(2x2), A=àI for All à € Real Number

In summary, the question is asking to prove that the only matrix that commutes with any other matrix is the identity matrix. One approach is to take two matrices, A and B, and solve for their values by multiplying AB = BA and considering all possible values for B. Another method is to use LaTeX code to display matrices, which can be done by using the package amsmath.
  • #1
annoymage
362
0
let A € M(2x2)..
if AB=BA for all B(2x2),
show that A=àI for all à € real number

its really hard to send new post using phone..
 
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  • #2
Do you understand the question? Basically, it's asking you to prove that the only matrix that commutes with any other matrix is the identity matrix.

Well, if you don't see any clever way to solve it (like me at first reading), you could simply take two matrices,

[tex]A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad B = \begin{pmatrix} x & y \\ u & v \end{pmatrix}[/tex]
and see what equations you get when you multiply AB = BA. Then note that this equations should hold for any values of x, y, u and v - allowing you to solve for a, b, c and d.
 
  • #3
before that,its hard to give my answer, so please teach me how to make that latex matrix.
 
  • #4
If you click on it, or quote my post, you can see the code I used.

(If you want to use pmatrix in your own LaTeX documents, you have to \usepackage{amsmath}).
 

1. How do you prove a linear algebra theorem?

To prove a linear algebra theorem, you must follow a logical and systematic approach. First, state the given information and what you need to prove. Then, use the definitions and properties of linear algebra to make deductions and draw conclusions. Finally, provide a clear and concise explanation of your reasoning to support your proof.

2. What is the importance of proving linear algebra theorems?

Proving linear algebra theorems is essential for understanding the fundamental concepts and principles of linear algebra. It allows us to verify the validity of mathematical statements and build a strong foundation for further studies in this field. Moreover, proofs help us to develop critical thinking and problem-solving skills that are valuable in many other areas of science and mathematics.

3. How do you know when a proof for a linear algebra theorem is correct?

A proof for a linear algebra theorem is considered correct if it follows a valid logical argument and is based on established definitions and properties. It should also be clear, concise, and easily understandable. Additionally, it is essential to check if the proof satisfies all the necessary conditions and if there are any counterexamples that can disprove it.

4. Can you use visual aids to prove a linear algebra theorem?

Yes, visual aids can be helpful in proving linear algebra theorems. Graphs, diagrams, and geometric representations can provide a better understanding of the concepts and relationships involved in the proof. They can also help to visualize the steps and logic used in the proof, making it easier to follow and validate.

5. Are there any shortcuts or tricks for proving linear algebra theorems?

While there are no shortcuts or tricks for proving linear algebra theorems, there are some techniques that can make the process more efficient. These include breaking down the proof into smaller, more manageable steps, using multiple methods to approach the problem, and looking for patterns or similarities with previous proofs. It also helps to have a good understanding of the fundamental concepts and properties of linear algebra.

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