  We continue to study the notion of cancellation-free linear circuits. We show
that every matrix can be computed by a cancellation- free circuit, and almost
all of these are at most a constant factor larger than the optimum linear
circuit that computes the matrix. It appears to be easier to prove statements
about the structure of cancellation-free linear circuits than for linear
circuits in general. We prove two nontrivial superlinear lower bounds. We show
that a cancellation-free linear circuit computing the $n\times n$ Sierpinski
gasket matrix must use at least 1/2 n logn gates, and that this is tight. This
supports a conjecture by Aaronson. Furthermore we show that a proof strategy
for proving lower bounds on monotone circuits can be almost directly converted
to prove lower bounds on cancellation-free linear circuits. We use this
together with a result from extremal graph theory due to Andreev to prove a
lower bound of {\Omega}(n^(2- \epsilon)) for infinitely many $n \times n$
matrices for every $\epsilon > 0$ for. These lower bounds for concrete matrices
are almost optimal since all matrices can be computed with $O(n^2/\log n)$
gates.
