Math 21b: Fact sheet about determinants

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]
For each n>3 there is an analogous formula, but you don't usually want to use it because it has lots of terms and the determinant can be computed more quickly using the properties below.

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).]

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(AT) = det(A) [6.2.7, page 266].