Why is the set of diagonal matrices not an ideal?

Question

Saw this fact in my lecture notes, but can't quite seem to work out why this is the case. Its clearly a subgroup of $M_n(\Re)$ and it seems that when multiplied by any element of $M_n(\Re)$ a diagonal matrix will result in a diagonal matrix. Definitely missing something insanely simple and trivial here but cant quite work it out.

Answer 1

What you're missing is that the product of a diagonal matrix with an arbitrary matrix is almost never a diagonal matrix. For instance, suppose that your diagonal matrix is the identity matrix and that the other matrix is any non-diagonal matrix.

Answer 2

Actually, when you left-multiply an $n\times n$ matrix by the diagonal matrix $D=\operatorname{diag}(a_1, a_2,\dots,a_n)$, for each $i=1,\dots, n$, you multiply row $i$ of the matrix by $a_i$, and when you right-mutiply it by $D$, it's column $i$ you multiply by $a_i$.