Explain why a 2x2 matrix can have at most two distinct eigenvalues
1 answer:
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
which correspond to the distinct Eigen values 
of an n * n matrix, then our set 
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 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.
So lets say that a 2 * 2 matrix (A) has three distinct eigenvalues.
You need to remember that the eigen vectors state
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
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An aritmetic sequence is like this
where a1=first term and d=common difference
geometric is
where a1=first term and r=common ratio
can it be both aritmetic and geometric
hmm, that means that the starting terms should be the same
therfor we need to solve
what values of d and r make all natural numbers of n true?
are there values that make all natural numbers for n true?
when n=1, then d(1-1)=0 and r^(1-1)=1, so already they are not equal
the answer is no, a sequence cannot be both aritmetic and geometric
geometric is
can it be both aritmetic and geometric
hmm, that means that the starting terms should be the same
therfor we need to solve
what values of d and r make all natural numbers of n true?
are there values that make all natural numbers for n true?
when n=1, then d(1-1)=0 and r^(1-1)=1, so already they are not equal
the answer is no, a sequence cannot be both aritmetic and geometric