# Matrix Binomials: Struggling to Interpret Question

• clarex
In summary, the conversation is discussing how to find the general statement for expressing M^n in terms of aX and bY. The suggestion is to diagonalize the matrix and take the nth power of the diagonal matrix, then apply the binomial theorem to expand the terms on the diagonal. Another interpretation is that (aX + bY)^n is sufficient.
clarex
Troubles interpreting what this wants from me.

X = (1 1
1 1)

Y = (1 -1
-1 1)

Let A = aX and B = bY, where a and b are constants.

Now consider M= (a+b a-b
a-b a+b)

Find THE general statement that expresses M^n in terms of aX and bY.

I'm completely lost. Can anyone direct me in the right direction?

Okay, so I'm sure you noticed that M = aX + bY. Then M^n = (aX + bY)^n, right? Now, if X and Y were just plain old real numbers (or variables if you like), what would you do to expand (aX + bY)^n

Especially since you titled this matrix binomials !

Diagonalize the matrix and then take the nth power of the diagonal matrix. You can generally extend scalar functions to matrix functions in this manner provided the eignvalues are unique and the eigenvectors or orthogonal. If you want once you have taken the nth power of the diagonal matrix you can apply the binomial theorm to expand the binomial terms on the diagonal.

My interpretation of "Find THE general statement that expresses M^n in terms of aX and bY" is that (aX+ bY)n would be sufficient.

## 1. What are matrix binomials?

Matrix binomials are mathematical expressions that involve matrices, which are rectangular arrays of numbers or symbols. They are typically written in the form (A + B)^n, where A and B are matrices and n is a positive integer.

## 2. What makes interpreting questions about matrix binomials challenging?

Interpreting questions about matrix binomials can be challenging because they involve complex mathematical concepts and operations. It requires a solid understanding of matrix algebra and the ability to manipulate matrices using various rules and properties.

## 3. How do I approach solving a question about matrix binomials?

The first step is to carefully read and understand the question. Next, identify the given matrices and any known values. Then, use the properties of matrix algebra to manipulate the expressions and solve for the unknown variables. It may also be helpful to draw diagrams or use examples to better visualize the problem.

## 4. What are some common mistakes to avoid when working with matrix binomials?

Some common mistakes when working with matrix binomials include not properly applying the rules of matrix algebra, incorrectly setting up the problem, and not carefully checking the final solution. It is important to double check all calculations and make sure they align with the given information and problem statement.

## 5. How can I improve my understanding of matrix binomials?

To improve your understanding of matrix binomials, it is important to practice solving a variety of problems and familiarize yourself with the properties and rules of matrix algebra. You can also seek help from a tutor or instructor, watch instructional videos, and read textbooks or online resources on matrix algebra. Additionally, it can be helpful to break down complex problems into smaller, more manageable steps and to constantly review and reinforce your knowledge through practice.

Replies
3
Views
2K
Replies
2
Views
2K
Replies
9
Views
4K
Replies
8
Views
1K
Replies
1
Views
843
Replies
12
Views
4K
Replies
5
Views
1K
Replies
4
Views
2K
Replies
8
Views
2K
Replies
2
Views
1K