You’ve made an error in computing the determinant: it should be $\lambda^3-4\lambda^2+5\lambda-2$. However, it’s not necessary to go through all of that for this particular matrix.
A matrix and its transpose have the same eigenvalues, so we can examine the rows of the matrix. From the last row, we have the left eigenvector $(0,0,1)$ with eigenvalue $1$. If you add the first and third rows, you get $(2,0,2)$, but left-multiplying a matrix by $(1,0,1)$ performs this addition, so we have another eigenvector/eigenvalue pair: $(1,0,1)$ and $2$. The trace of a matrix is equal to the sum of its eigenvalues, so we can find the last eigenvalue “for free:” it’s $4-2-1=1$. With a bit of practice, you’ll be able to spot things like this.