A is a 5 times matrix with eigenvalues 1 3 and 8 Null A I i

A is a 5 times matrix with eigenvalues -1, 3, and 8. Null (A + I) is a plane, Null (A + 8I) is also a plane. Is A diagonalizable?

Solution

Let us suppose that A is any matrix of order 5x5 with eigenvalues -1, 3 and 8.

Since Null (A + I ) represents a plane, it means that A has two linearly indepndent eigenvectors corresponding to eigenvalue r = -1. ( because for a plane we need two vectors )

Also,   Null (A - 8I ) represents a plane, it also means that A has two linearly indepndent eigenvectors corresponding to eigenvalue r = 8.

And, for each eigenvalue there must be at least one eigenvector, so A has one eigenvector for eigenvalue r = 3, therefore matrix A has 5 linearly independent eigenvectors for three distinct eigenvalues.

Hence, A is diagonalizable.