Problem:
Consider systems of three linear equations with unknowns , and ,
where each of the coefficients is either or and the system has a solution other than . For example, one such system is
with a nonzero solution of . How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
First consider systems in which one or more of the equations is . There is such system in which all three equations are . If exactly one of the equations is , then the other two equations can be any of possibilities and can occur in any of the 3 positions in the system, for possibilities. Similarly, if exactly two of the equations are , then the other equation can be any of 7 possibilities and can occur in any of the 3 positions in the system, for possibilities.
From now on assume that none of the equations is . There are possibilities in which all three equations are the same, and possibilities in which exactly two are the same.
From now on assume further that all three equations are different. If one of the three variables is missing from all the equations, then, up to a permutation of variables or equations, the system must be ; there are systems of this type. Otherwise, all three variables are present and up to a permutation of variables or equations the system must be , and there are systems of this type. The requested total is .
Note: The question is equivalent to asking how many matrices all of whose entries are 0 or 1 have determinant 0 . The corresponding answer for systems of four homogeneous linear equations with four unknowns (or matrices) is 42,976 . In each case, the fraction of systems with a nontrivial solution is to the nearest whole percent.
The problems on this page are the property of the MAA's American Mathematics Competitions