This is a consequence of what's called Cayley-Hamilton Theorem
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In general, for an nxn matrix M, write det(M-λI) as an n-degree polynomial in λ then let λ=M and set it to 0
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What about the generalized nxn matrix? Does this relationship only apply to 2x2, or is it also satisfied by higher order matrices
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for higher order the coefficients are a bit nastier but do have a closed form
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This is a specific case of the more general form. For a characteristic polynomial p(x) = a_n x^n + ... + a_1 x + a_0*id, a_n = (-1)^n, a_{n-1} = -Trace(A), a_0 = (-1)^n det(A), all others can be written as traces of powers of A but it’s really messy in closed form
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See thread
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I learned this at school but still don't have an intuitive understanding of what it means
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On real numbers, it is the equivalent to say that the solutions of the polynomial X^2-S×X+P=0 are always such that their sum is S and their product is P
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