Question

Determine whether the statement The homogenous equation Ax = least one free variable.

below is true or false. Justify the answer. 0 has the trivial solution if and only if the equation has at.
Choose the correct answer below.
A. The statement is true. The homogeneous equation Ax = 0 has the trivial solution if and only if the matrix A has a row of zeros which implies the equation has at least one free variable.
B. The statement is false. The homogeneous equation Ax = 0 never has the trivial
solution.
C. The statement is true. The only if the equation has homogenous equation Ax = 0 has the trivial solution if and
at least one free variable which implies that the equation has a nontrivial solution.
D. The statement is false. The homogeneous equation Ax = 0 always has the trivial
solution.

Answer 1

The statement is false since every homogeneous equation Ax = 0 always has the trivial solution, regardless of the presence of free variables.

Step-by-step explanation:

The statement 'The homogenous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable' is false. The correct justification is that a homogeneous equation Ax=0 always has the trivial solution, which is the zero vector. This is because if you multiply the matrix A by the zero vector, you'll get the zero vector, satisfying the equation Ax=0 for any matrix A. A free variable in a system represents a dimension of the solution space where infinite solutions exist, which does not affect the existence of the trivial solution. Therefore, the existence of free variables is related to the presence of non-trivial solutions, not the trivial one.