$*$Matlab$*$

Now try an easier approach. Calculate the eigenvectors and eigenvalues of A using the command $[$P lambda $]=$ eig$($A$),$ and check that all resulting eigenvectors are linearly independant $($Hint: caluclate the determinant of $P)$:

Now you can easily see what the operation A does: it stretches the component of the vector in the $(1,1,1)$ direction by a factor of $2$ and leaves the component perpedicuar to $(1,1,1)$ as is$.$

Calculate the component of $r_1$ in the direction of $(1,1,1)$ and the component perpendicular to $(1,1,1)[$Hint: the scalar or dot product of $2$ vectors can be calculated with the command dot$(r1,r2).$