Linear Algebra - homogeneous equation

[Inertigratus]

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?

 

[statdad]

When you plug in the known matrices $A^2, aA, bI_s$ you will get a $2 \times 2$ matrix on the left side: it has four entries. On the right you have the matrix

$$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

Write down the left matrix and right matrix: what do you see?

 

[Inertigratus]

Isn't it 4 x 3?
13 + 3a + b
5 + a
4 + 4a
8 - 2a + b
on the left side?

 

[statdad]

You have four equations on the left - how many constants are on the right?

 

[Inertigratus]

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.

 

[statdad]

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 $2 \times 2$ 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.