The problem, stated here, was the following:

what is an exact expression for the singularity probability $f(p)$ of a $3\times 3$ random matrix with iid Rademacher entries having the distribution $P(1) = p, P(-1) = 1-p$, to find its average over $p$, and to show that it is minimized at $p = \frac{1}{2}$.

Student members Yixin Wang (Columbia University) and Tengyuan Liang (Wharton School, University of Pennsylvania), provided correct solutions to the various parts of the problem.

*Yixin Wang (left) and Tengyuan Liang both answered correctly*

This can actually be done by a brute force complete enumeration. Alternatively, condition on the first two rows and denote the third row by $(x,y,z)$, and calculate the determinant. It is a linear function of $x, y, z$; find the probability that the determinant is zero. Now uncondition and obtain the unconditional probability that the determinant is zero. At first sight, the singularity probability $f(p)$ is a polynomial of degree $9$. But cancellation occurs and it is actually the sixth degree

polynomial

\[ f(p) = 1-18p^2+84p^3-162p^4+144p^5-48p^6.\]

Immediately, the average singularity probability is

$\int_0^1 f(p)\,dp = \frac{26}{35}$, and $f(\frac{1}{2}) = \frac{5}{8}$.

Pictures can be deceiving, and this is an instance. A coarse scale picture would suggest that $f(p)$ is convex over the unit interval. It is not. It has the concave-convex-concave shape.

$f”(p)$ is a polynomial of degree four and has a double zero at $p = \frac{1}{2}$ and two other distinct zeros in the unit interval. It has no other real or complex roots. The first derivative $f'(p)$ is a polynomial of degree five, has a zero at $p = 0, 1,$ and $\frac{1}{2}$, and the zero at $\frac{1}{2}$ is a triple zero. The first derivative has no other roots. From the symmetry of $f(p)$ and the fact that it is strictly decreasing in a right neighborhood of zero and so, strictly increasing in a left neighborhood of $p = 1$, it now follows that the minimum must be at $p = \frac{1}{2}$.

We have here the function $f(p)$ written explicitly; an interesting question is whether the coefficients in the polynomial have some relations to a well known number theoretic function. The same question could be asked for other values of $n$.

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