What scalar value, computed from the elements of a square matrix, indicates whether the matrix is invertible?

What scalar value, computed from the elements of a square matrix, indicates whether the matrix is invertible?

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

Answer: (A)

The determinant is the scalar value that tells you whether a square matrix is invertible. It measures how the associated linear transformation scales volumes. If the determinant is nonzero, the transformation preserves a volume (up to a scale) and an inverse transformation exists. If the determinant is zero, the transformation collapses space to a lower dimension, so no inverse exists. This makes the determinant the direct indicator of invertibility: zero means not invertible, nonzero means invertible. Other measures like rank describe how many independent rows or columns exist (full rank implies invertibility, but rank is not a single scalar computed from the entries in the same straightforward way as the determinant). The trace sums diagonal entries and provides no general rule about invertibility, and eigenvalues relate to invertibility only via whether zero is among them, which is less immediate than checking the determinant.

The determinant is the scalar value that tells you whether a square matrix is invertible. It measures how the associated linear transformation scales volumes. If the determinant is nonzero, the transformation preserves a volume (up to a scale) and an inverse transformation exists. If the determinant is zero, the transformation collapses space to a lower dimension, so no inverse exists.

This makes the determinant the direct indicator of invertibility: zero means not invertible, nonzero means invertible. Other measures like rank describe how many independent rows or columns exist (full rank implies invertibility, but rank is not a single scalar computed from the entries in the same straightforward way as the determinant). The trace sums diagonal entries and provides no general rule about invertibility, and eigenvalues relate to invertibility only via whether zero is among them, which is less immediate than checking the determinant.