In matrix notation, what does the expression |A| denote?

In matrix notation, what does the expression |A| denote?

• A. trace
• B. rank
• C. Determinant
• D. eigenvalue

Answer: (C)

The bars around a square matrix denote its determinant, a single number that captures how the linear transformation scales volumes. It also tells you about invertibility: if the determinant is zero, the transformation collapses some volume and the matrix has no inverse; if it’s nonzero, the transform is invertible. The determinant has useful rules, like det(AB) = det(A) det(B) and det(A^T) = det(A). For an n×n matrix, det(cA) = c^n det(A). The determinant also equals the product of the eigenvalues (counting multiplicities). By contrast, the trace is the sum of diagonal entries, the rank is the dimension of the column (or row) space, and eigenvalues come from the characteristic equation det(A − λI) = 0. So the expression denotes the determinant.

The bars around a square matrix denote its determinant, a single number that captures how the linear transformation scales volumes. It also tells you about invertibility: if the determinant is zero, the transformation collapses some volume and the matrix has no inverse; if it’s nonzero, the transform is invertible. The determinant has useful rules, like det(AB) = det(A) det(B) and det(A^T) = det(A). For an n×n matrix, det(cA) = c^n det(A). The determinant also equals the product of the eigenvalues (counting multiplicities). By contrast, the trace is the sum of diagonal entries, the rank is the dimension of the column (or row) space, and eigenvalues come from the characteristic equation det(A − λI) = 0. So the expression denotes the determinant.