A matrix is considered invertible when which of the following conditions is met?

A matrix is considered invertible if its determinant does not equal zero. This is foundational in linear algebra. The determinant provides crucial information about a matrix's properties, specifically regarding whether it has an inverse.

When the determinant of a matrix is non-zero, it implies that the matrix is of full rank, which means that its columns (or rows) are linearly independent. Linear independence is essential because it ensures that the matrix transforms vectors in a way that maintains distinctness, allowing an inverse transformation to exist.

In contrast, if the determinant is zero, the matrix loses this independence, meaning it maps some vectors to the same point, hence creating an inability to reverse the transformation uniquely. Thus, a zero determinant indicates that the matrix is singular and does not possess an inverse.

The other conditions listed do not relate directly to the invertibility of a matrix. The trace being zero or the determinant being exactly one does not provide sufficient information regarding the ability to invert the matrix. Similarly, the sign of eigenvalues does not directly correlate to whether a matrix is invertible. Therefore, the requirement that the determinant be non-zero is the sole condition that confirms a matrix's invertibility.