examining if a matrix is diagonizable

Question

I was practicing some linear algebra problems and I stopped at this one:

Without calculating the eigenvectors, show that the following matrix is diagonalizable and find the diagonal matrix to which it is similar.

$$ \begin{bmatrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \\ \end{bmatrix} $$

How could I know the if its diagonalizable or not without calculating the eigenvectors?
Please help, I'm studying for tomorrow's exam.

Answer 1

Hint - It is a $\;3\times 3\;$ matrix with $\;3\;$ different eigenvalues

Answer 2

The characteristic polynomial is $(x-3)(x-2)(x-5)$ as you can see directly from the matrix. This is also the minimal polynomial. A matrix is diagonalizabe iff its minimal polynomial is a product of distinct linear factors. The diagonalization has the same diagonal as the original matrix.

Answer 3

The given matrix is an upper triangular.We know that determinant of an upper triangular matrix is the product of its diagonals.Thus clearly characteristic polynomial is $(x-3)(x-2)(x-5)$ and so it has $3$ distinct eigen values.Now if a $n\times n$ matrix has $n$ distinct eigen values then it is diagonalisable.

Hence given matrix is diagonalisable.