Error while Diagonalizing the Matrix

I'm studying State Space function thesedays, and made a code below

A = [ 0 1 0; 0 0 1; -6 -11 -6];
P = [ 1 1 1; -1 -2 -3; 1 4 9 ];
S = P^(-1);
A1 = S*A*P;

Expected answer was

A1 = [ -1 0 0 ; 0 -2 0 ; 0 0 -3 ]

because of the eigenvalue process, however, it made a conclusion as

A1 = [ -1, -1.7764e-15, -2.6645e-15 ; 2.6645e-15, -2, 7.1054e-15 ; -1.3323e-15, -2.6645e-15, -3 ]

And other variables are calculated just as I expected.

I don't know what makes this kind of error and how can I solve it?

It can be a critical error in the process.

[Mehmed Saad]

round(A1,14)

Answer 1

[Walter Roberson]

Do not use P^(-1), which is also known as inv(P), if you can avoid doing so. The calculations have too much numeric difficulty.

Use the \ or / operator insted.

A1 = P\A*P

The answer will not be the exact integers you are expecting, but it will be closer to what you are expecting.

If you need exact values then do not work numerically, work symbolically:

A1 = P\sym(A)*P