Linear Algebra, 5th Edition

4.2 Determinants of Order n

In this section, we extend the definition of the determinant to $n\times n$ matrices for $n\ge 3.$ For this definition, it is convenient to introduce the following notation: Given $A\in {\text{M}}_{n\times n}(F),$ for $n\ge 2,$ denote the $(n-1)\times (n-1)$ matrix obtained from A by deleting row i and column j by ${\tilde{A}}_{ij}.$ Thus for

$$A=\left(\begin{array}{lll}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)\in {\text{M}}_{3\times 3}(R),$$

we have

$${\tilde{A}}_{11}=\left(\begin{array}{ll}5& 6\\ 8& 9\end{array}\right),{\tilde{A}}_{13}=\left(\begin{array}{ll}4& 5\\ 7& 8\end{array}\right),\text{and}{\tilde{A}}_{32}=\left(\begin{array}{ll}1& 3\\ 4& 6\end{array}\right),$$

and for

$$B=\left(\begin{array}{rrrr}1& -1& 2& -1\\ -3& 4& 1& -1\\ 2& -5& -3& 8\\ -2& 6& -4& 1\end{array}\right)\in {\text{M}}_{4\times 4}(R),$$

we have

$${\tilde{B}}_{23}=\left(\begin{array}{rrr}1& -1& -1\\ 2& -5& 8\\ -2& 6& 1\end{array}\right)\text{and}{\tilde{B}}_{42}=\left(\begin{array}{rrr}1& 2& -1\\ -3& 1& -1\\ 2& -3& 8\end{array}\right).$$

Definitions.

Let $A\in {\text{M}}_{n\times n}(F).$ If $n=1,$ so that $A=(A_{11}),$ we define $\det (A)=A_{11}.$ For $n\ge 2,$ we define det(A) recursively as

$$\det (A)={\displaystyle \sum _{j=1}^n{(-1)}^{1+j}{A}_{1j}\cdot \det ({\tilde{A}}_{1j})}.$$

The scalar det(A) is called the determinant of A and is also denoted by $\left|A\right|.$ The scalar

$${(-1)}^{i+j}\det ({\tilde{A}}_{ij})$$

is called the ...