Linear Algebra, 5th Edition

4.4 Summary—Important Facts about Determinants

In this section, we summarize the important properties of the determinant needed for the remainder of the text. The results contained in this section have been derived in Sections 4.2 and 4.3; consequently, the facts presented here are stated without proofs.

The determinant of an $n\times n$ matrix A having entries from a field F is a scalar in F, denoted by det(A) or $\left|A\right|$, and can be computed as follows.

1. If A is $1\times 1$, then $\text{det}(A)=A_{11},$ the single entry of A.

2. If A is $2\times 2,$ then $\text{det}(A)=A_{11}{A}_{22}-A_{12}{A}_{21}$. For example,

   $$\text{det}\left(\begin{array}{rr}-1& 2\\ 5& 3\end{array}\right)=(-1)(3)-(2)(5)=-13.$$

3. If A is $n\times n$ for $n>2,$ then, for each i, we can evaluate the determinant by cofactor expansion along row i as

   $$\text{det}(A)={\displaystyle \sum _{j=1}^n{(-1)}^{i+j}}{A}_{ij}·\text{det}({\tilde{A}}_{ij}),$$

   or, for each j, we can evaluate ...