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Mark each statement as true or false. Suppose A is an n n matrix. a. If an n × n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable. False b. If A is diagonalizable, then A is also diagonalizable False c. If there is a basis of R n consisting of eigenvectors of A, then A is diagonalizable. True d. A is diagonalizable if and only if A has n eigenvalues, counting multiplicity. False e. If A is diagonalizable, then A is invertible. False

Respuesta :

The correct option for the matrix will be:

  • False
  • True
  • True
  • False
  • False

How to explain the matrix?

a) If an n x n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.

  • False

It could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue.

b) If A is diagonalizable the A2 is diagonalizable

  • True

If A is diagonalizable then there exists an invertible matrix

c) If Rn has a basis of eigenvectors of A, then A is diagonalizable.

  • True

d) A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.

  • False

e) If A is diagonalizable, then A is invertible.

  • False

It’s invertible if it doesn’t have a zero as eigenvalue but this doesn’t affect diagonalizable.

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