Which scalar is computed from a square matrix and is zero if and only if the matrix is singular?

Which scalar is computed from a square matrix and is zero if and only if the matrix is singular?

• A. Determinant of a Matrix
• B. Trace
• C. Rank
• D. Eigenvalue

Answer: (A)

The main idea tested is how the determinant relates to whether a square matrix is invertible. The determinant is a scalar that tells you how the associated linear transformation scales volume. If you apply the matrix to a unit cube, the resulting parallelepiped’s volume equals the determinant. When the determinant is nonzero, the transformation preserves a nonzero volume and the matrix is invertible. When the determinant is zero, the image collapses to a lower-dimensional object, volume becomes zero, and the matrix is singular. This is exactly the zero-for-singular relationship. The other options don’t fit this precise criterion: the trace is the sum of diagonal entries and can be zero even for invertible matrices; rank describes how many independent rows or columns there are and is zero only for the zero matrix; eigenvalues are a set of scalars (the matrix has multiple eigenvalues) and while having zero as an eigenvalue indicates singularity, there isn’t a single universal scalar that is guaranteed to be zero exactly for all singular matrices. The determinant uniquely provides that zero-or-not criterion for singularity.

The main idea tested is how the determinant relates to whether a square matrix is invertible. The determinant is a scalar that tells you how the associated linear transformation scales volume. If you apply the matrix to a unit cube, the resulting parallelepiped’s volume equals the determinant. When the determinant is nonzero, the transformation preserves a nonzero volume and the matrix is invertible. When the determinant is zero, the image collapses to a lower-dimensional object, volume becomes zero, and the matrix is singular.

This is exactly the zero-for-singular relationship. The other options don’t fit this precise criterion: the trace is the sum of diagonal entries and can be zero even for invertible matrices; rank describes how many independent rows or columns there are and is zero only for the zero matrix; eigenvalues are a set of scalars (the matrix has multiple eigenvalues) and while having zero as an eigenvalue indicates singularity, there isn’t a single universal scalar that is guaranteed to be zero exactly for all singular matrices. The determinant uniquely provides that zero-or-not criterion for singularity.