Chapter11

If a matrix is symmetric, then it is:

• Never diagonalizable

• Always diagonalizable

• Sometimes diagonalizable

• Diagonal only if eigenvalues are distinct

According to the Diagonalization Theorem, an n x n matrix A is diagonalizable if and only if which of the following conditions is met?

• The matrix A must be a symmetric matrix.

• The matrix A must have n linearly independent eigenvectors.

• The matrix A must have n distinct eigenvalues.

• The determinant of A must be non-zero.

• A is not a diagonal matrix

• The only eigenvalue of A is λ = 4

• The eigenvalues of A are λ = 4 and λ = 1

• A is not diagonalizable

• Yes, because it is triangular.

• No, because it has only one eigenvector

• Yes, because the determinant is non-zero

• No, because it has negative entries

• 𝐴 is not diagonalizable because it has a repeated eigenvalue.

• 𝐴 is diagonalizable because it has three linearly independent eigenvectors.

• 𝐴 is not diagonalizable because one eigenvalue is negative.

• 𝐴 is diagonalizable only if all entries of the matrix are distinct.

Find the distinct eigenvalues of the matrix A shown below

• -4,-4,6,6

• 4,4,-6,-6

• -4,6

• 4,-6

• The matrix A must have n linearly independent eigenvectors

• The matrix A must be invertible

• The matrix A must be symmetric

• det(A) must be non zero

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