# Given #M = ((1, 1, 1), (0, 5, 5), (0, 0, 7))#, is it true that there is a non-zero second degree polynomial of which #M# is a root?

It is false.

We can check directly as follows:

Given:

We find:

Then:

Subtracting the second equation from the third we find:

Then from the second equation:

Then from the first equation:

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Yes, it is true. Since the matrix M has a non-zero determinant (det(M) = 35), it implies that there exists a non-zero second-degree polynomial for which M is a root, based on the Cayley-Hamilton theorem.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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