If a linear transformation matrix applied to a vector space has an eigenvalue of zero, which outcome is inevitable?

💡 Explanation

A zero eigenvalue implies that the linear transformation maps some non-zero vector to the zero vector; this means the null space is non-trivial, and therefore, the determinant is zero, because the matrix is singular and not invertible. Therefore, option C is correct, rather than a dimension change or orthogonality, because a zero eigenvalue directly impacts invertibility.

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