Is the last principal component the minimum-variance portfolio?

Three assets have annual return covariance matrix (in $\%^2$)

You are told its eigenvectors are $(1, 2, 2)/3$, $(2, 1, -2)/3$ and $(2, -2, 1)/3$.

A colleague says: "The last principal component is the lowest-variance direction, so rescale it to sum to 1 and you have the fully invested minimum-variance portfolio."

• Find the eigenvalues and the share of variance each principal component explains.
• Compute the colleague's portfolio and its variance.
• Find the true minimum-variance portfolio with $w_1 + w_2 + w_3 = 1$ and its variance.
• Under what condition on $\Sigma$ would the colleague be right?