After extensive testing with glasses of water, I’ve determined that it’s impossible to stably balance one infinitely thin circle on top of another. This is true whether the circles are the same size or different, and does not depend on the mass distribution of the circles. This is because two circles intersect at either zero or two points (unless they are the same, and that still isn’t a stable configuration).

If $a$ is a scalar and $M$ is a square matrix, it is very convenient to be able to write $a + M$. Usually people know immediately what this means, but are uneasy about “abusing” notation, so here’s the detailed justification for why this is perfectly legitimate:

Matrices should be considered first and foremost as linear transformations. You know what a matrix is if you know what it does to vectors. A scalar is also a linear transform on vectors: multiplying a scalar times a vector is a linear operation. Therefore, scalars can also be thought of as linear transformations, and therefore as matrices. It is immediate which matrix the scalar should be: the result of multiplying by a scalar $k$ is that all components are scaled by $k$; the matrix that does this is just $kI$.