What is the quickest yet systematic way to solve this question?
Consider the system of linear equations:
$$w + 3x + 2y +2z = 0$$
$$w + 4x + y = 0$$
$$3w + 5x + 10y + 4z = 0$$
$$2w+ 5x + 5y + 6z = 0$$
with solutions of the form $(w,x,y,z)$, where $w,x,y,z$ are real. Which of the following statements is false:
A. The system is consistent.
B. The system has infinitely many solutions.
C. The sum of any two solutions is a solution.
D. $(-5,1,1,0)$ is a solution.
E. Every solution is a scalar multiple of $(-5,1,1,0)$
A related question is if we have a large $(4\times 4$ or $5\times 5$) matrix, and we can't tell if the rows/columns are linearly independent just by looking at them, how do we tell if the matrix is invertible or not?
Best Answer
Start with (D). Obviously, it's true. This also means that the determinant of this homogeneous system is zero.
Jump to (A). The system has at least one solution, so it's consistent. In fact all homogeneous systems are consistent because a vector of zeros is always a solution.
Jump to (B). Multiply the solution from (D) with $k$ and you will see that $(-5k,k,k,0)$ is also a solution. The number of solutions is infinite.
Jump to (C). If you have two different solutions $X,Y$ such that $AX=0,AY=0$, then $A(X+Y)=0$. So a linear combination of solutions is also a solution.
Jump to (E). You'll have to prove that the matrix rank is 3 (by Gaussian elimination, for example). With a 4x4 matrix it should be fairly simple exercise. This proves the fact that (E) is actually true.