The determinant of any square matrix A is a scalar, denoted det(A). [Non-square matrices do not have determinants.]
The determinant of a square matrix A detects whether A is invertible:
If det(A)=0 then A is not invertible (equivalently, the rows of A are linearly dependent; equivalently, the columns of A are linearly dependent);
If det(A) is not zero then A is invertible (equivalently, the rows of A are linearly independent; equivalently, the columns of A are linearly independent).
[Fact 6.2.2, page 263]
In particular, if any row or column of A is zero then det(A)=0; if two rows or two columns are proportional, then again det(A)=0.
Formulas for determinants of n-by-n matrices when n is small:
Laplace expansion by minors down a column or across a row: express the determinant of an n-by-n matrix in terms of n determinants of (n-1) by (n-1) matrices [6.1.4 and 6.1.5, pages 252 and 253].
Examples of easy Laplace expansions when A is "sparse" (has lots of well-placed zero entries) [pages 252-253].
In particular: the determinant of an upper or lower triangular matrix
is the product of its diagonal entries [6.1.6, page 253].
Special case: the determinant of an identity matrix In always equals 1.
The determinant is not a linear function of all the entries
(once we're past the boring case of n=1).
But if we fix all the entries of A except
one row or one column v then det(A) is
linear as a function of v. [6.2.8, page 267]
[Application: the determinant of the scalar multiple cA of an n-by-n matrix A is cndet(A).]