Question

6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let 2 be an eigenvalue of multiplicity 3. If A-2/ has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).

Answer 1

Let   be a  $5\times 5$ matrix with 3 different eigen values. One of them has multiplicity 3, then the other two have 1.

Let $\lambda_1$ has multiplicity 3 and  $\lambda_2,\lambda_3$ has multiplicity 1.

Now for $\lambda_2,\lambda_3$ we will definitely get 1 eigen vector each that is total 2.

and now we have,

$A-\lambda_1I$ has rank 2. Then the number of eigen vector for $\lambda_1$ is 2.

Therefore, the total number of eigen vector is 4 (<5)

Hence, The matrix is defective.