Which of the following conditions is necessary for a matrix to be invertible?

For a matrix to be invertible, a fundamental condition that must be satisfied is that its determinant must not be zero. When a matrix has a non-zero determinant, it indicates that the matrix is full rank, meaning that its columns (or rows) are linearly independent. This linear independence is critical because it ensures that there are no solutions to the homogeneous equation (Ax = 0) other than the trivial solution (x = 0), which signifies the uniqueness of solutions to the equation (Ax = b) for any vector (b).

If the determinant is zero, the matrix is singular, implying that it does not span the entire space and lacks an inverse. This condition confirms that the rows or columns of the matrix are linearly dependent, leading to infinitely many solutions or no solutions at all for certain linear systems. Thus, having a non-zero determinant is essential for constructing the inverse of a matrix, making this condition sufficient to guarantee that the matrix can indeed be inverted.

In summary, the requirement that the determinant must not be zero is what ensures the matrix's invertibility, aligning with the necessary linear algebra concepts related to matrix theory.