Help with a simple matrix proof

In summary, the conversation discusses how to prove that if A and B commute with C, then AB also commutes with C. This can be done by manipulating the formula for matrix multiplication, as it is associative.
  • #1
paulrb
18
1
Edit: Sorry, I figured this out shortly after posting. It's a simple problem but I hadn't used matrix equations before.

Homework Statement



If A, B, and C are all square matrices of the same size, show that AB commutes with C if A and B both commute with C.

Homework Equations



Formula for matrix multiplication

The Attempt at a Solution



All I know is
AC = CA
BC = CB
A, B, and C are square matrices of the same size

I want to prove that (AB)(C) = (C)(AB).
But I am not sure how to do this, or even how to get started.
 
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  • #2
Well Matrix multiplication is associative so you should be able to manipulate (AB)C to get C(AB) pretty easily.
 

1. What is a matrix proof?

A matrix proof is a mathematical technique used to prove statements or theorems involving matrices. It involves manipulating and analyzing matrices using various rules and properties to show that a statement is true.

2. How do I start a simple matrix proof?

To start a simple matrix proof, you first need to state the theorem or statement that you want to prove. Then, you can begin by manipulating the matrices using basic operations such as addition, subtraction, and multiplication to reach the desired result.

3. What are some key properties and rules used in matrix proofs?

Some key properties and rules used in matrix proofs include the associative, commutative, and distributive properties, as well as the properties of matrix multiplication, inverse matrices, and identity matrices. Additionally, the properties of determinants and matrix rank can also be used in certain proofs.

4. Are there any common mistakes to avoid when doing a matrix proof?

Yes, there are a few common mistakes to avoid when doing a matrix proof. These include forgetting to use parentheses when necessary, incorrect matrix multiplication or addition, and assuming that certain properties hold true when they may not be applicable in the given scenario. It is important to carefully check all steps in the proof for accuracy.

5. How can I improve my skills in doing matrix proofs?

To improve your skills in doing matrix proofs, it is important to have a strong understanding of basic matrix operations and properties. Practice is also key, so solving a variety of different matrix proofs and seeking help when needed can also be beneficial. Additionally, studying and understanding various proof techniques and strategies can help in tackling more complex matrix proofs.

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