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 $A={\displaystyle {\sum}_{j=1}^{n}{(-1)}^{1+j}{a}_{1j}{m}_{1j}=}{\displaystyle {\sum}_{j=1}^{n}{a}_{1j}{c}_{1j}}$

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

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