# Check whether the matrices A and B are diagonalisable?Diagonalise those matrices which are diagonalisable. (i) #A={[-2,-5,-1],[3,6,1],[-2,-3,1]}# (ii) #B={[-1,-3,0],[2,4,0],[-1,-1,2]}#.

##### 1 Answer

#### Explanation:

For the matrix

After some algebra this can be recast in the form

or

Thus the eigenvalues are 1,2 and 2.

Since the eigenvalue 2 is repeated (it is algebraically degenerate) there is a possibility that there may not be enough independent eigenvectors to form a diagonalizing matrix. To check that, let us try to find the independent eigenvector(s) corresponding to

By elementary row operations (or just by appropriately adding or subtracting the equations) we get

Since **not** diagonalizable.

It is easy to see that

Here the eigenvector(s) satisfy

It is obvious that in this case the only constraint on the eigenvectors is

To complete the diagonalization, we need to find the eigenvector corresponding to

So, a diagonalizing matrix is