# Answer on Linear Algebra Question for AA

Question #1379

(a) Let A be a square matrix and fA(x) (x = lambda) its characteristic polynomial. In each of the following cases (i) to (iv), write down whether

(D) A is diagonalizable over R

(N) A is not diagonalizable over R

(U) it is not possible to say one way or the other.

(i) fA(x) = (x - 3)^2(x - 5)

(ii) fA(x) = (x^2 - 1)(x^2 - 2)

(iii) fA(x) = (x^2 + 6)(x - 1)(X + 2)

(iv) fA(x) = (x^2 + 3)^2

(b) For each case that you marked (N), say whether or not the matrix can

certainly be diagonalized over C

(D) A is diagonalizable over R

(N) A is not diagonalizable over R

(U) it is not possible to say one way or the other.

(i) fA(x) = (x - 3)^2(x - 5)

(ii) fA(x) = (x^2 - 1)(x^2 - 2)

(iii) fA(x) = (x^2 + 6)(x - 1)(X + 2)

(iv) fA(x) = (x^2 + 3)^2

(b) For each case that you marked (N), say whether or not the matrix can

certainly be diagonalized over C

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