If (zero matrix) has only the trivial solution then has a unique solution for all choices of in .

Understanding the Problem

The statement refers to linear algebra concepts, particularly regarding solutions to linear systems represented by matrices. A zero matrix is a matrix in which all the elements are zero. The claim suggests that if the zero matrix has only the trivial solution, it implies there is a unique solution for all other choices of the variables in the system.

Relevant Concepts

1. Zero Matrix: A matrix where all elements are zero.
2. Trivial Solution: In linear algebra, the trivial solution refers to the solution where all variables are set to zero.
3. Linear Independence: A set of vectors (or equations) is linearly independent if there are no non-trivial combinations of them that yield the zero vector (or zero matrix).
4. Unique Solution: A system of equations has a unique solution if it has exactly one solution, which occurs when the number of equations matches the number of variables (and they are independent).

Analysis and Detail

1. Zero Matrix Case: If the equation $A \vec{x} = 0$ only has the trivial solution $\vec{x} = \vec{0}$, it means that the matrix $A$ has full rank, implying its columns are linearly independent.
2. Implication on Solutions: If $A$ has full rank, then for any non-homogeneous equation $A \vec{x} = \vec{b}$, where $\vec{b} \neq 0$, the system will have a unique solution as long as $\vec{b}$ is in the column space of $A$.

Verify and Summarize

If a zero matrix resulting from $A \vec{x} = 0$ has only the trivial solution, it confirms that plenty of factors such as linear independence, rank condition, and the matrix structure allows for unique solutions across varying $\vec{b}$. Therefore, the statement holds under these considerations.

Final Answer

If the zero matrix has only the trivial solution, it implies that the matrix has full rank, ensuring a unique solution for all choices of the variables in the equation $A \vec{x} = \vec{b}$.