Let M be an n×n matrix such that M2−3M+2I is equal to 0, where I is the n×n identity matrix. Is M diagonalizable?

Answer 1

Yes, M is diagonalizable.

Proof: Let the matrix M be a matrix consisting of polynomials of characteristic 2 where I is the identity matrix such that,

M2-3M+2I = X²-3X+2.

The polynomial is factor-able in M such that X²-3X+2 = (X-1)(X-2).

A square matrix is diagonalizable if (and only if) its minimal polynomial factors completely into distinct linear factors over the base field F. The factors X-1  and X-2 are distinct over any field (even a field of characteristic 2). Thus, M is diagonalizable.