Question

True or False? Circle your answer and justify your choice.

(a) T F : If A is a 13 × 17 matrix over R, then AT A is diagonalizable.
(b) T F : A 3 × 3 matrix A with only one eigenvalue cannot be diagonalizable.
(c) T F : A matrix A of size n×n is diagonalizable if and only if A has n distinct eigenvalues. g

Answer 1

a and c are true

Explanation:

The question is about diagonalizability of a matrix.

a) Given that A is a 13x17 matrix over R

We get
`A^T A` will be a square matrix 13x13.

Being a square matrix this is diagonalizable provided eigen values are independent.

True

b) A 3 × 3 matrix A with only one eigenvalue cannot be diagonalizable.

False. For a square matrix to be diagonalizable, the eigen values should be independent.

c) A matrix A of size n×n is diagonalizable if and only if A has n distinct eigenvalues. g

This is true, because for a square matrix with distinct eigen values we can see that it is diagonalizable.