Matrix with entities 0 and 1

Question

The number of $3\times 3$ matrix A whose entities are either 0 or 1 and for which the system $A\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ 0\\ 0 \end{array}} \right]$ has exactly two distinct solution is

(A) 0

(B) $2^9-1$

(C) 168

(D) 2

How do we proceed this type of problem I have asked a similar type of problem Symmetrical Matrices with entities $0$ and $1$ but how do we do this problem as the matrix may or may not be symmetric.

Over which field do you do the matrix multiplication? The answers may be different for $\Bbb F_2, \Bbb F_3, \Bbb R$ etc... — Jonas Linssen, Dec 16 '20 at 10:51

score 0 · Answer 1 · answered Dec 18 '20 at 03:56

Obviously the correct choice is (A)

Three situations can only occur while solving a system of three linear equations in three variables:

(i) No solution(the three planes form a triangular prism i.e line of intersection of the two planes is parallel to the third plane or at least two of the three planes are parallel)

(ii) Exactly one solution(the three planes intersect in a point)

(iii) Infinite number of solutions(the three planes intersect in a line or the three equations represent one plane only) .

So there cannot be any matrix $A$ which creates system of linear equations having exactly two solutions

Matrix with entities 0 and 1

1 Answers1