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:

• n=1: det([a]) = a [see 6.1.4, p.251]
• n=2: ad-bc [p.247; also seen in earlier chapters]
• n=3: box product of columns; Sarrus rule [6.1.1 and 6.1.2, page 248]

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 I_n 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).]

Further properties:

• Behavior under elementary row operations [6.2.1, page 262]; computation of det(A) by row reduction [6.2.3, page 263].
• The determinant is multiplicative: for any square matrices A,B of the same size we have
  det(AB) = (det(A)) (det(B)) [6.2.4, page 264]. The next two properties follow from this.
• The determinant of the inverse of an invertible matrix is the inverse of the determinant:
  det(A^-1) = 1 / det(A) [6.2.6, page 265].
• Similar matrices have the same determinant; that is, if S is invertible and of the same size as A then det(S A S^-1) = det(A).
  [6.2.5, page 265. In other words, the determinant of a linear transformation from Rⁿ to itself remains the same if we use different coordinates for Rⁿ.]
  Finally,
• The determinant of the transpose of any square matrix is the same as the determinant of the original matrix:
  det(A^T) = det(A) [6.2.7, page 266].