Proving properties of matrices

I don't have the knowledge to do the proof.In summary, the conversation discusses the commutativity of matrices A and B, and how this property can be used to prove that CD equals DC for matrices C and D defined as A^2 + 2*A and A^3 + 5 * I, respectively. This can be generalized to any arbitrary matrices. The conversation then moves on to discussing the relationship between rotation matrices and how this can be used to derive formulas for cos(theta + phi) and sin(theta + phi) in terms of cos(theta), cos(phi), sin(theta), and sin(phi). The speaker mentions not having enough knowledge to provide a proof for this concept.
  • #1
Soluz
2
0
1) I can assume all these matrices to be 2x2.
We have matrix A and B and AB = BA, that is, they commute.. Prove if C = A^2 + 2*A and D = A^3 + 5 * I (I is identity matrix), then CD = DC.
Then give a theory that generalizes this.

2) why does R(theta)R(phi)=R(theta+phi)? (explain with "simple" words)
Knowing this, derive the formulas cos(theta + phi) and sin(theta + phi) in terms or cos(theta),cos(phi),sin(theta), and sin(phi).

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I haven't done a proof course yet so I'm completely lost.
 
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  • #2
Take 2 arbitrary matrices [arbitrary values] and check what happens to these arbitrary values if the 2 matrices hold AB=BA.
Then apply it.
 

1. What is the purpose of proving properties of matrices?

The purpose of proving properties of matrices is to establish mathematical rules and relationships that can be used to manipulate matrices and solve problems involving them. These properties can also help in understanding the properties of other mathematical objects that can be represented using matrices.

2. What are some commonly used properties of matrices?

Some commonly used properties of matrices include the commutative property, associative property, distributive property, and the inverse property. These properties can be used to simplify calculations and solve equations involving matrices.

3. How do you prove properties of matrices?

To prove properties of matrices, one must use logical reasoning and mathematical operations to show that the property holds true for any given set of matrices. This may involve using algebraic manipulations, properties of matrix operations, or mathematical proofs.

4. Why is it important to prove properties of matrices?

Proving properties of matrices helps to ensure the accuracy and validity of mathematical calculations involving matrices. It also allows for a deeper understanding of the relationship between matrices and other mathematical concepts, and can provide insights into new ways of solving problems.

5. Can properties of matrices be applied to real-world problems?

Yes, properties of matrices can be applied to real-world problems in fields such as engineering, physics, and computer science. Matrices are often used to represent data and perform calculations, and knowing their properties can help in finding efficient and accurate solutions to real-world problems.

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