In a linear transformation represented by a matrix, what conclusion can be made if the transformation results in the same output for different inputs?

When a linear transformation represented by a matrix yields the same output for different inputs, this indicates that the transformation is not injective, meaning it does not uniquely map each input to a distinct output. This scenario occurs when there is at least one pair of distinct inputs that map to the same output, suggesting that the transformation collapses multiple inputs into a single output.

For a matrix to produce such behavior, it implies that the columns of the matrix are linearly dependent. Linear dependence in the context of a matrix means that there exists a non-trivial combination of its columns that results in the zero vector. This dependence is mathematically reflected in the determinant of the matrix being zero, indicating that the matrix does not have full rank and thus cannot span the entire output space.

Since the determinant of a matrix is zero if and only if the columns are linearly dependent, the conclusion that the matrix has a zero determinant is valid. This is the reason why the answer indicates that the matrix indeed has a zero determinant in cases where the transformation yields the same output for different inputs. It aligns with the core principles of linear algebra regarding the properties of linear transformations and their corresponding matrices.