Chapter 7 - Eigenvalues and Eigenvectors - 7.2 Diagonalization - 7.2 Exercises - Page 360: 26

We can guarantee the diagonalization.

We know that for a triangular or diagonal matrix, the eigenvalues are the entries on the diagonal. Hence, here the eigenvalues are: $4,1,-2$. So we have $3$ distinct eigenvalues and the dimensions of the matrix are $3\times3$. Thus, we can guarantee the diagonalization.