<Dd> p (t) I n = t n I n + t n − 1 c n − 1 I n + ⋯ + t c 1 I n + c 0 I n, (\ displaystyle p (t) I_ (n) = t ^ (n) I_ (n) + t ^ (n - 1) c_ (n - 1) I_ (n) + \ cdots + tc_ (1) I_ (n) + c_ (0) I_ (n) ~,) </Dd> <P> one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients . </P> <P> Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t is the same on both sides; it follows that the constant matrices with coefficient t in both expressions must be equal . Writing these equations then for i from n down to 0, one finds </P> <Dl> <Dd> B n − 1 = I n, B i − 1 − A B i = c i I n for 1 ≤ i ≤ n − 1, − A B 0 = c 0 I n . (\ displaystyle B_ (n - 1) = I_ (n), \ qquad B_ (i - 1) - AB_ (i) = c_ (i) I_ (n) \ quad (\ text (for)) 1 \ leq i \ leq n - 1, \ qquad - AB_ (0) = c_ (0) I_ (n) ~ .) </Dd> </Dl>

Prove that every square matrix is a root of its characteristic polynomial