Let B = a b c d be an arbitrary matrix in M2×2(F). (a) Use the formula for the determinant of a 2×2 matrix to give an explicit formula for the characteristic polynomial of B. (b) Show that B2 ∈ tr(B)B...

Let B =

a b
c d
be an arbitrary matrix in M2×2(F).
(a) Use the formula for the determinant of a 2×2 matrix to give an explicit formula for the characteristic polynomial of B.
(b) Show that B2 ∈ tr(B)B + det(B)I2 = 0, where 0 denotes the zero matrix. (Hint: Use the Cayley-Hamilton Theorem.)