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Mathematics for Physics

245 Determinant

A scalar attached to a linear transformation that is used to measure signed volume scaling of a basis under that transformation.

definition [d] (Determinant) From Axler: the determinant of \(T\), denoted \(\det T\), is defined to be the unique number in \(F\) such that

  • \(\alpha T = (\det T)\, \alpha\)

for all \(\alpha \in \Lambda^{\dim V}(V^{*})\), written in Axler’s notation as all \(\alpha\) in the space of alternating forms of degree \(\dim V\).

where

  • \(T\) is an operator on a finite-dimensional vector space \(V\) over a field \(F\).
  • \(\det T\) is the determinant of \(T\).
  • \(\alpha\) is an alternating form of top degree on \(V\).

245.1 Elementary Example

245.1.1 Simple

For a \(2 \times 2\) matrix, the determinant is the familiar product difference of entries.

\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

\[ \det A = ad - bc \]

\[ A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix},\quad \det A = 6 \]

where

  • \(a, b, c, d\) are scalars.
  • \(\det A\) is the determinant of \(A\).

245.1.2 General

For a \(3 \times 3\) diagonal matrix, the determinant is the product of the diagonal entries.

\[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \]

\[ \det A = 2 \cdot 3 \cdot 5 = 30 \]

where

  • \(\det A\) equals the product of the eigenvalues when \(A\) is diagonal.