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My book defines the characteristic equation of A and says it can be expanded to "polynomial form":

|λI - A| = λn + cn-1λn-1 + ... + c1λ + c0

Can someone explain to me how they got this polynomial? Wouldn't it need to include a bunch of cofactors, or something? And why is lambda raised to the nth power? And where did the constants come from? And where did any value of A go...I'm just confused.

Or is my book just defining the determinant to be equal to that, and we're not actually supposed to understand where they got it from?

Edit: Could someone show me a step-by-step calculation of how they got this, or something, so that I can really understand?

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    A *step-by-step* calculation of the thing you ask is indeed possible, but it mostly relies on your ability to manipulate the notation $\sum\prod\sum$. For instance, how do you feel about this formula? $$\det(\lambda I-A)=\sum_{\sigma\in\mathfrak S_n}\prod_{i=1}^n (\lambda\delta_{i,\sigma(i)}-a_{i,\sigma(i)})$$ Because this is the starting point, if you want to eventually come up with $c_0,\cdots,c_{n-1}$ in terms of $a_{i,j}$. – 2017-02-14

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No matter what, the determinant of a matrix has, as at least one term, the product of all elements along its diagonal. This means there is a product of the form $$\prod_{i=1}^n (\lambda - a_{ii})$$ somewhere in the determinant, which can be expanded as an nth-degree polynomial in $\lambda$. Therefore, $\det(\lambda I - A)$ is overall an nth-degree polynomial in $\lambda$.

EDIT: to clarify, they're not giving a formula for each coefficient, as it'd be about the same as doing the calculation. But the determinant will be of that form.

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    I don't really get why at least one term has the product of all elements along its diagonal? – 2017-02-14
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    Refer to the 2D determinant, ad - bc. The "ad" is the product on the diagonal. In general, the determinant is the sum of all signed products of elements where, in each product, only one element from each row and from each column is present. The diagonal has to be one of these terms. – 2017-02-14