Just needed to check these since I dont particularly enjoy linear algebra, and im a little rusty.
(1) If I have an n*n matrix that has n different eigenvalues, that automatically makes the matrix diagonalizable, does it not?
(2) If an n*n matrix has n eigenvalues, some of which share the same value, can it still be diagonalizable?
Matlab gives eigenvalues
Remark the fourth one is very small. It may be 0. That would mean the determinant is 0 and the matrix would not be invertible.
Note the second and 6th rows that have zeros in the same place.
So they're good candidates to find a linear combination.
There's already 1 at the same place. Remain 19 and 12 in the 8th column.
Find a row that will eliminate the difference between 19 and 12, namely row 5.
row2 = row6 + (7/4) row5
hence the rows are not linearly independent and hence the matrix is not invertible.
EDIT: I did not realize the symmetry, but flyingsquirrel realized it!