Kunze Linear Algebra · Problem 5.2.13
Determinants
Mathematics — Methods · multiplicative-determinant-functions
Problem
Let be the field of real numbers, and let be a function on matrices over , with values in , such that for all , . Suppose also that Prove the following. ((a)) ; ((b)) if ; ((c)) if is obtained by interchanging the rows (or columns) of ; ((d)) if one row (or one column) of is ; ((e)) whenever is singular.Intuitive explanation
The key insight is that is a multiplicative function on matrices—it respects the product structure—so it must behave like a determinant. The condition that distinguishes the identity from a row-swap matrix tells us is not trivial, which forces it to capture something essential about matrix structure. Use the multiplicative property strategically: if you can write a matrix as a product involving nilpotent or singular factors, or express it in terms of matrices whose -values you know, you can deduce of the original. The path to each part involves finding clever factorizations or relationships that leverage .
Intuitive explanation
The key insight is that is a multiplicative function on matrices—it respects the product structure—so it must behave like a determinant. The condition that distinguishes the identity from a row-swap matrix tells us is not trivial, which forces it to capture something essential about matrix structure. Use the multiplicative property strategically: if you can write a matrix as a product involving nilpotent or singular factors, or express it in terms of matrices whose -values you know, you can deduce of the original. The path to each part involves finding clever factorizations or relationships that leverage .