Does Every Linear System's Structure Predict Its Number of Solutions?

[baher]

1)Determine whether the given statement is true or false.
A homogeneous linear system with the same number of equations as unknowns always
has a unique solution.

2)Determine whether the given statement is true or false.
If a linear system has no solution, the rank of the coecient matrix must be less than
the number of equations.

3)Determine whether the given statement is true or false.
If a linear system has the same number of equations as unknowns and the coecient
matrix is invertible, the system has exactly one solution.

My answers was true false true , is that right ? :D

 

[chiro]

Hey baher and welcome to the forums.

For these kind of questions, it helps the other readers and members here to understand what you are thinking and how you arrived at the particular answers you arrived at.

I will start by commenting on the first question.

Consider that you have a system with n-equations with n-unknowns. This is a square matrix. This means it has a determinant. What values does a determinant have? What happens when that determinant is 0?