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Linear Algebra - homogeneous equation

  • #1
128
0

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:

Answers and Replies

  • #2
statdad
Homework Helper
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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
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Isn't it 4 x 3?
13 + 3a + b
5 + a
4 + 4a
8 - 2a + b
on the left side?
 
  • #4
statdad
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You have four equations on the left - how many constants are on the right?
 
  • #5
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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
statdad
Homework Helper
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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.
 

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