# Algebraic/Matrix Manipulation (linear algebra)

• boings
In summary, the conversation discusses the property that a matrix must have in order to obtain non-trivial solutions for a given equation. It is mentioned that a matrix must be invertible and have a non-zero determinant. The trivial solution, x=0, is also mentioned and it is noted that the determinant must be 0 for there to be other solutions.
boings

## Homework Statement

I have attached the relevant question as an image (for sake of ease)

## The Attempt at a Solution

Also attached, in blue.

Thanks a lot for any help at all!

#### Attachments

• Matrixmanip.jpg
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You could consider ##B^k Y = 0##. Which property does B need to get non-trivial solutions for Y? How is that related to a?
##Y=(B-I_2)X##. How do you get Y=0 (which is a solution to the equation above) with non-trivial X? How is that related to a?

Perhaps the very beginning is where I'm having trouble, I don't know what times a matrix would be equal to zero. Is it the transpose? Or perhaps a matrix whose determinant is zero?

If you have studied matrices at all then you should know this basic property: the equation Ax= y has a unique solution if and only if A is invertible: $x= A^{-1}y$. And that is only true if A has non-zero determinant.

The equation Ax= 0 always has the "trivial" solution, x= 0. It has other solutions if and only if its determinant is 0.

You're right, I should know that basic property :) I've been cramming too much this semester so I tend to forget.

Thanks a lot it makes good sense.

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the manipulation of matrices, vectors, and other mathematical objects to solve problems related to systems of linear equations, transformations, and more.

## 2. What are the applications of linear algebra?

Linear algebra has a wide range of applications in fields such as physics, engineering, computer science, economics, and statistics. It is used to solve problems related to data analysis, computer graphics, optimization, signal processing, and many other areas.

## 3. What are matrices and how are they manipulated?

Matrices are rectangular arrays of numbers or other mathematical objects. They are manipulated using various operations such as addition, subtraction, multiplication, and division. These operations follow specific rules and can be used to solve systems of linear equations, find inverse matrices, and perform other calculations.

## 4. What is the difference between algebraic and matrix manipulation?

Algebraic manipulation involves solving equations using algebraic operations such as addition, subtraction, multiplication, and division. Matrix manipulation, on the other hand, involves working with matrices and performing operations such as matrix multiplication, finding determinants, and calculating eigenvalues and eigenvectors.

## 5. How is linear algebra used in machine learning?

Linear algebra is a fundamental tool in machine learning and artificial intelligence. It is used to represent and manipulate data in the form of vectors and matrices, perform calculations and transformations, and build algorithms for tasks such as pattern recognition, classification, and prediction.

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