Under what condition does a square matrix have an inverse?

Under what condition does a square matrix have an inverse?

• A. det M ≠ 0
• B. det M = 0
• C. M is symmetric
• D. M is diagonalizable

Answer: (A)

A square matrix is invertible if and only if its determinant is not zero. This is because a nonzero determinant means the linear transformation represented by the matrix scales volumes by a nonzero factor, so no information is lost and there exists an inverse transformation that undoes it. Conversely, if the determinant is zero, the transformation collapses some dimension, the rows (or columns) are linearly dependent, and no matrix can reverse that loss of information, so no inverse exists. Properties like symmetry or diagonalizability don’t guarantee invertibility on their own: a symmetric matrix can have det zero, and a diagonalizable matrix can be singular if zero is one of its eigenvalues.

A square matrix is invertible if and only if its determinant is not zero. This is because a nonzero determinant means the linear transformation represented by the matrix scales volumes by a nonzero factor, so no information is lost and there exists an inverse transformation that undoes it. Conversely, if the determinant is zero, the transformation collapses some dimension, the rows (or columns) are linearly dependent, and no matrix can reverse that loss of information, so no inverse exists. Properties like symmetry or diagonalizability don’t guarantee invertibility on their own: a symmetric matrix can have det zero, and a diagonalizable matrix can be singular if zero is one of its eigenvalues.