Question about the Smith normal form of a matrix over the field of integers

Question

I understand that I would need to perform elementary row and column operations. So for a matrix
$M = \begin{array}{cc} -1 & 1 \\ 0 & 2 \\ \end{array}$

am I correct in saying that it cannot be reduced further than $M = \begin{array}{cc} 1 & 0 \\ 0 & 2 \\ \end{array}$?
Same thing with any diagonal matrix say $M = \begin{array}{cc} 2 & 0 \\ 0 & 5 \\ \end{array}$ to begin with, it cannot be reduced further, right? Appreciate any hint. I was just introduced to normal forms.

Answer 1

The Smith normal form of $\pmatrix{2&0\\0&5}$ is $\pmatrix{1&0\\0&10}$. More generally, the SNF of $\pmatrix{a&0\\0&b}$ is $\pmatrix{\gcd(a,b)&0\\0&\text{lcm}(a,b)}$.

With $\pmatrix{2&0\\0&5}$ you can add column $2$ to column $1$ to get $\pmatrix{2&0\\5&5}$ and then you can reduce the first column via row operations to $\pmatrix{1\\0}$. The second column will then be $\pmatrix{*\\\pm10}$ and one finally gets to $\pmatrix{1&0\\0&10}$.