Question

10. Which of the following is not a necessary condition for a matrix, say A, to be diagonalizable?

a. A must have n linearly independent eigen vectors

b. All the eigen values of A must be distinct

c. A can be an idempotent matrix

d. A must have n linearly dependent eigen vectors

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Answer 1

Answer:

A can be an idempotent matrix... right answer....

Answer 2

The correct answer is option (d) A must have n linearly dependent eigen vectors.

Explanation:

• A must have n linearly dependent eigen vectors is not a necessary condition for a matrix, say A, to be diagonalizable.
• The theorem of diagonalization tells us that, ‘An n×n matrix A is diagonalizable, if and only if, A has n linearly independent eigenvectors.’
• So, if A has n distinct eigen values, say λ1, λ2, λ3…λn, then the corresponding eigen vectors are said to be linearly independent.
• One thing more, all idempotent matrices are said to be diagonalizable.
• Hence, the correct answer is option d i.e A must have n linearly dependent eigen vectors.