Which of the followings are true or false. Justify your answers

a. The solution set to a system of three equations in three unknowns cannot be a plane.
b. A system of linear equations cannot have only two solutions.
c. The solution set to a consistent rank 2 linear system in four unknowns would be a line in four-dimensional space.
d. A system of four equations in four unknowns always has a solution.
e. A system of four equations in four unknowns can have at most one solution

[tex]x-y-z=1\\x+2y-2z=2\\3x+3y-3=3\\\\\begin{pmatrix} 1 & 1 & -1|&1\\ 2&2&-2|&2\\3&3&-3|&3\end{pmatrix}\\\\\\R_2 \rightarrow R_2-R_1\\R_3 \rightarrow R_3-R_1\\\\\begin{pmatrix} 1 & 1 & -1|&1\\ 0&0&~~0|&0\\0&0&~~0|&0\end{pmatrix}\\\\x+y-z=1[/tex]

thus there is a plane

b. If a system of solution has more than one solution than it can have infinite amount of solutions so there can be any amount of solutions except 2.

c. The solution set of a consistent rank 2 linear system with four unknowns will be a plane in the four dimensional space.

d. Following is an example which proves it wrong i-e no solution

[tex]\\\\\\\\x_1=1\\x_2+x_3=2\\x_2+x_3+x_4=0\\[/tex]

e.as shown from following example it has infinite solutions

[tex]\\\\\\\\x_1+x_2=1\\2x_1+2x_2=2\\x_3=1\\x_4=1\\[/tex]

so a system can have more than one solution