Which statement about a non-singular matrix is true?

Which statement about a non-singular matrix is true?

• A. It is always symmetric
• B. It has determinant zero
• C. It is always diagonalizable
• D. It has an inverse

Answer: (D)

A non-singular matrix is one that has a nonzero determinant, which is exactly what guarantees it can be inverted. Because the determinant is not zero, there exists a matrix that serves as its inverse, so the statement about having an inverse is true. The other ideas don’t have to hold in general: a matrix can be non-singular and still not be symmetric, for example a simple upper triangular matrix with nonzero diagonal entries is invertible but not symmetric. Saying the determinant is zero would make it singular, so that contradicts non-singularity. And diagonalizability isn’t guaranteed either—there are non-singular matrices that are not diagonalizable, such as a Jordan block with eigenvalue 1, which has determinant 1 but cannot be put into a diagonal form. Therefore, the correct property is that a non-singular matrix has an inverse.

A non-singular matrix is one that has a nonzero determinant, which is exactly what guarantees it can be inverted. Because the determinant is not zero, there exists a matrix that serves as its inverse, so the statement about having an inverse is true. The other ideas don’t have to hold in general: a matrix can be non-singular and still not be symmetric, for example a simple upper triangular matrix with nonzero diagonal entries is invertible but not symmetric. Saying the determinant is zero would make it singular, so that contradicts non-singularity. And diagonalizability isn’t guaranteed either—there are non-singular matrices that are not diagonalizable, such as a Jordan block with eigenvalue 1, which has determinant 1 but cannot be put into a diagonal form. Therefore, the correct property is that a non-singular matrix has an inverse.