Multiplication of projection matrices Let $A$ and $B$ be two projection matrices having same dimensions. Then does the following hold
\begin{equation}
AB\leq I,
\end{equation}
where $I$ is the identity matrix. In other words is it true that $I-AB$ is positive semi-definite.

It's not clear to me what you mean by matrix inequality, but assuming you mean an inequality componentwise, it's clearly false, for instance let
$$A = B = \frac{1}{2} \left[\begin{array}{cc}1 & 1\\1 &1\end{array}\right]$$
be projection onto the vector $(1,1)$.
Then $AB = A$ has positive off-diagonal entries so violates your inequality.
EDIT: $(I-AB)$ is not positive semi-definite, as projection matrices are not in general symmetric. For instance
$$A = \left[\begin{array}{cc}1 & 2\\0 & 0\end{array}\right]$$
is a projection matrix ($A^2 = A$) but $I - A^2 = I-A$ is clearly not symmetric.
