Properties of Determinants

Here we collect the basic properties of determinants with brief justifications. They are used to simplify computations (see the methods in the article Determinant) and in theory. The notions of minor and cofactor that we rely on are defined in the article Determinant.

Throughout, $A$ is a square matrix of order $n$ .

1. Determinant of a Triangular (Diagonal) Matrix

The determinant of a triangular or diagonal matrix equals the product of the entries on the main diagonal:

a_{11} 0 ⋮ 0 a_{12} a_{22} 0 \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{nn} = a_{11} a_{22} \dots a_{nn} .

In particular, for the identity matrix $det E = 1$ .

(Proved by successively expanding along the first column.)

2. Invariance Under Transposition

det (A^{T}) = det A .

So any property that holds for the rows of a determinant also holds for the columns (and vice versa). Below we state properties for rows, with columns implied.

3. Swapping Two Rows

If two rows (columns) are swapped, the determinant changes sign:

det A^{'} = - det A .

4. Two Identical Rows

If a determinant has two identical rows (columns), then $det A = 0$ .

Proof. Swap these rows. On the one hand, by property 3 the determinant becomes $- det A$ ; on the other hand, the matrix has not changed, so it still equals $det A$ . From $det A = - det A$ it follows that $det A = 0$ .

5. Pulling Out a Common Factor

A common factor of the entries of a row (column) can be pulled out of the determinant:

λ a_{11} a_{21} ⋮ λ a_{12} a_{22} \dots \dots = λ a_{11} a_{21} ⋮ a_{12} a_{22} \dots \dots .

Corollary. When the entire matrix is multiplied by a scalar $λ$ , the factor is pulled out of each of the $n$ rows:

det (λ A) = λ^{n} det A .

6. Proportional Rows

If two rows (columns) are proportional, then $det A = 0$ (pulling out the proportionality coefficient by property 5 gives two identical rows — property 4).

7. Zero Row

If all the entries of some row (column) are zero, then $det A = 0$ (this is the case $λ = 0$ in property 5; alternatively, expand along that row).

8. Linearity in a Row

If every entry of some row is written as a sum of two terms, then the determinant equals the sum of two determinants, in which that row consists of the first and second terms respectively, with all other rows unchanged:

a_{11} + b_{11} a_{21} a_{12} + b_{12} a_{22} = a_{11} a_{21} a_{12} a_{22} + b_{11} a_{21} b_{12} a_{22} .

9. Adding a Multiple of One Row to Another

If the entries of another row (column), multiplied by the same number, are added to the entries of one row (column), the determinant does not change.

Proof. By property 8 the new determinant splits into the original one plus a determinant with two proportional rows, which equals zero (property 6).

This is precisely the property underlying the order-reduction and triangular-reduction methods for computing determinants.

10. Determinant of a Product

det (A \cdot B) = det A \cdot det B (A, B are square, of the same order) .

Corollary. For a nonsingular matrix, $det (A^{- 1}) = \frac{1}{det A}$ (since $det (A^{- 1}) det A = det E = 1$ ).

11. Linearly Dependent Rows

If one of the rows (columns) is a linear combination of the others, then $det A = 0$ . This generalizes properties 6 and 7; it is equivalent to the fact that the determinant is zero if and only if the rows of the matrix are linearly dependent (that is, the matrix is singular) — see Linear Dependence and Independence of Rows.

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