(a) Prove that for any 2 × 2 matrix T there are scalars c0, . . . , c4 that are not all 0 such that the combination c4T4 + c3T3 + c2T2 + c1T + c0I is the zero matrix (where I is the 2 × 2 identity matrix, with 1's in its 1, 1 and 2, 2 entries and zeroes elsewhere, see Exercise 34).
(b) Let p(x) be a polynomial p(x) = cnxn + . . . + c1x + c0. If T is a square matrix we define p(T) to be the matrix cnTn + . . . + c1T + c0I (where I is the appropriately-sized identity matrix). Prove that for any square matrix there is a polynomial such that p(T) is the zero matrix.
(c) The minimal polynomial m(x) of a square matrix is the polynomial of least degree, and with leading coefficient 1, such that m(T) is the zero matrix. Find the minimal polynomial of this matrix.

(This is the representation with respect to ε2, ε2, the standard basis, of a rotation through π/6 radians counterclockwise.)

Transcribed Image Text:

V3/2 1/2 1/2 B/2,