Linear Algebra - homogeneous equation

In summary, The problem is to find for which a's and b's the equation A2 + aA + bI2 = 0 is true, using a linear system and gaussian elimination. The equation can be split into four equations with respect to the variables a and b and when the constants are moved to the right side, there will be four equations in two unknowns.
  • #1
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Homework Statement


The problem is setting up the equation, it says that the matrix equation will be made up of four equations for the 2 unknowns.
I'm supposed to find for which a's and b's the equation is true, using a linear system and gaussian elimination.

Homework Equations


A2 + aA + bI2 = 0
A = | 3 1|
...| 4 -2|
I2 = identity matrix, 2x2
A2 = | 13 5 |
...| 4 8 |

The Attempt at a Solution


I'm not sure how to proceed with this problem.
How do I split the equation into 4 equations with respect to the variables a and b?
 
Last edited:
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  • #2
When you plug in the known matrices [itex] A^2, aA, bI_s [/itex] you will get a [itex] 2 \times 2 [/itex] matrix on the left side: it has four entries. On the right you have the matrix

[tex]
\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
[/tex]

Write down the left matrix and right matrix: what do you see?
 
  • #3
Isn't it 4 x 3?
13 + 3a + b
5 + a
4 + 4a
8 - 2a + b
on the left side?
 
  • #4
You have four equations on the left - how many constants are on the right?
 
  • #5
4? since all equation are equal to zero?
I guess I could move the constans to the right side. Then I would have four rows x three columns.
 
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  • #6
Inertigratus said:
4? since all equation are equal to zero?
I guess I could move the constans to the right side. Then I would have four rows x three columns.

You have 4 equations, one for each entry in the [itex] 2 \times 2 [/itex] matrix.
When you move all constants to the right you end up with four equations in two unknowns. I have not worked through the solution so I can't tell you what to expect when you move on to solving for the unknowns.
 

What is a homogeneous equation in linear algebra?

A homogeneous equation in linear algebra is an equation where all the terms have a coefficient of zero. In other words, there are no constant terms in the equation. The solution to a homogeneous equation is called the trivial solution, which means that all variables are equal to zero.

What are the properties of homogeneous equations?

Homogeneous equations have the following properties:

  • The zero vector is always a solution.
  • If the equation is consistent (has a solution), it has infinitely many solutions.
  • The solutions form a vector space.
  • The equation can be solved using Gaussian elimination.
  • Homogeneous equations are used to find the null space of a matrix.

What is the relationship between homogeneous equations and linear independence?

A set of vectors is linearly independent if and only if the only solution to the homogeneous equation formed by those vectors is the trivial solution (all variables equal to zero). Therefore, homogeneous equations can be used to test for linear independence.

How are homogeneous equations used in solving systems of linear equations?

Homogeneous equations can be used to find the null space of a matrix, which is the set of all solutions to the corresponding homogeneous equation. This is helpful in solving systems of linear equations, as the null space can provide information about the solutions to the system.

What are some real-world applications of homogeneous equations in linear algebra?

Homogeneous equations are used in various fields, including physics, engineering, and economics. They can be used to model systems with no external influences, such as a pendulum swinging in a vacuum or a chemical reaction in a closed system. In economics, homogeneous equations can be used to model supply and demand curves without any external factors affecting the market. Additionally, homogeneous equations are used in image processing and computer graphics to transform images and objects.

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