Chapter 4

# Determinants and Eigenvalues

Luz M. DeAlba

Drake University

Throughout this chapter, F denotes an arbitrary field. All matrices are assumed to be in Fn × n, unless otherwise stated. A vector space V is assumed to be a vector space over a field F.

## 4.1 Determinants

Definitions:

The determinant, det A, of a matrix A = [aij] is an element in F defined inductively:

• det [a] = a.
• For i, j ∈ {1, 2,... ,n}, the ijth minor of A corresponding to aij is defined by mi,j = det A ({i}, {j}), where A ({i}, {j}) denotes the submatrix of A obtained by deleting the ith row and the jth column.
• The ijth cofactor of aij is cij = (−1)i+jmi,j.
• det $A={\sum }_{j=1}^{n}{\left(-1\right)}^{1+j}{a}_{1j}{m}_{1j}={\sum }_{j=1}^{n}{a}_{1j}{c}_{1j}$

This method of computing the determinant of a matrix is called Laplace expansion ...

Get Handbook of Linear Algebra, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.