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DivineSolar
Hey there. Hope I can help.

So lets say that a 2 * 2 matrix (A) has three distinct eigenvalues.

You need to remember that the eigen vectors state [tex]V_1,...,v_r[/tex] which correspond to the distinct Eigen values [tex]λ_1,...,λ_n[/tex] of an n * n matrix, then our set [tex]{V_1,...,v_r}[/tex] is linearly independent. 

Which we can now tell by this theorem that three linearly independent eigenvectors correspond which is literally absurd.

The reason for this is because the matrix A is only two dimensional which means the three vectors belong to R^2. So any set [tex]{V_1,...,v_r}[/tex] in R^n would be linearly independent if r > n since (r = 3) > (n = 2). These three eigenvectors then become linearly independent. Therefore the 2 * 2 matrix can only have 2 atmost.
facundo3141592

We want to explain why a 2x2 matrix can have at most two distinct eigenvalues.

Let's see this with the direct calculations.

Remember that for a matrix A the eigenvalues are given by:

[tex]det[(A - I*\lambda) = 0[/tex]

Where I is the identity matrix, which only has ones in the diagonal.

So the maximum number of eigenvalues λ is defined by the size of the matrix, for example in a 2x2 matrix, the corresponding identity matrix will have only two elements on the diagonal.

Thus, the maximum number of possible different eigenvalues is 2.

If you want to learn more, you can read:

https://brainly.com/question/13987127

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