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Explain why a $2 \times 2$ matrix can have at most two distinct eigenvalues. Explain why an $n \times n$ matrix can have at most $n$ distinct eigenvalues.

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Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

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on N by N Matrix. So you have a 11 and then you have a lemon. And similarly, you have B and limb and e and and everything that was in between. Okay. Uh OK, so these are not These are that's just your key. And then you have lander times I and you're subtracting this room, which is just lambda all the way up to London and times that them zero zero, everything else is you. Okay, so this matrix is just a 11 minus lambda, and then you have 812 even. And and then you have a one. A tow minus. Lambda it and I should have bigger bracket. But I think you get the idea and then eight and minus one. Okay. And you're solving you take the determinant of this and million matrix, and then you're solving for Lambda. Okay, so when you dated, return on what is the leading home? The leading dome or the term of the highest degree? Okay, so you would take product off, so you will take product off. All these storms died totems. They will come in your characteristic for a normal, and then you're solving it trade. So term off. The highest degree will be minus one to the power and Langer. And okay, plus all the other times. So you're characteristic polynomial is off Big end. Okay. And a degree of polynomial can have at most end this thing routes That's for never Did he remove algebra? Okay, a degree. And paranormal cannot have more than investing groups. And these rules are basically your Eigen values. So when so when n million metrics cannot have more than end distant back and values, OK, And if you taken, do we do, then you to weigh two matrix can have at most two distinct taken village, OK?

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