Determinant

A determinant is a numerical characteristic of a square matrix that makes it possible to solve many problems in linear algebra (invertibility of a matrix, solvability of systems, areas and volumes). Geometrically, the determinant can be understood as the "scaling factor" by which a linear transformation changes area (volume).

The determinant of a matrix $A$ is denoted $det A$ or $∣ A ∣$ .

Remark. A rigorous definition of a determinant of order $n$ is given via permutations (as an algebraic sum of $n!$ signed products) — see the article Determinant via Permutations. Here we use the equivalent recursive definition: a determinant of order $n$ is expressed through determinants of order $n - 1$ (expansion along a row or column).

Determinants of Small Orders

Order 1. $det (a_{11}) = a_{11}$ .

Order 2:

det A = a_{11} a_{21} a_{12} a_{22} = a_{11} a_{22} - a_{12} a_{21} .

Example: $3214 = 3 \cdot 4 - 1 \cdot 2 = 10.$

Minor and Cofactor

These notions are needed for expanding a determinant.

Definition. The minor $M_{ij}$ of the entry $a_{ij}$ is the determinant of order $n - 1$ obtained from the matrix $A$ by deleting row $i$ and column $j$ .

Definition. The cofactor $A_{ij}$ of the entry $a_{ij}$ is its minor, taken with the sign $(- 1)^{i + j}$ :

A_{ij} = (- 1)^{i + j} M_{ij} .

That is, $A_{ij} = M_{ij}$ if $i + j$ is even, and $A_{ij} = - M_{ij}$ if $i + j$ is odd.

Example. For

102 - 2 43 3 - 1 1

the minor of the entry $a_{23} = - 1$ is $M_{23} = 12 - 2 3 = 3 - (- 4) = 7$ , and the cofactor is $A_{23} = (- 1)^{2 + 3} M_{23} = - 7$ .

Third-Order Determinant

The rule of triangles. A $3 \times 3$ determinant is the algebraic sum of six products: three with a " $+$ " sign (the main diagonal and two "triangles" parallel to it) and three with a " $-$ " sign (the anti-diagonal and two "triangles" parallel to it):

a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33} = a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{13} a_{22} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} .

Expansion along a row (column). The same determinant can be computed by expanding along the first row (using cofactors):

det A = a_{11} a_{22} a_{32} a_{23} a_{33} - a_{12} a_{21} a_{31} a_{23} a_{33} + a_{13} a_{21} a_{31} a_{22} a_{32} .

Worked Example

A = 247358169 .

Expand along the first row:

det A = 25869 - 34769 + 14758 .

5869 = - 3, 4769 = - 6, 4758 = - 3.

det A = 2 \cdot (- 3) - 3 \cdot (- 6) + 1 \cdot (- 3) = - 6 + 18 - 3 = 9.

Expansion of a Determinant of Arbitrary Order

Theorem (11 (Laplace expansion)). A determinant equals the sum of the products of the entries of any row (or any column) with their cofactors. For example, expansion along row $i$ :

det A = a_{i 1} A_{i 1} + a_{i 2} A_{i 2} + \dots + a_{in} A_{in} = j = 1 \sum n a_{ij} A_{ij},

and expansion along column $j$ :

det A = i = 1 \sum n a_{ij} A_{ij} .

It is convenient to expand along the row (or column) with the most zeros. An extreme case:

Corollary. If all the entries of a row (column) except one, $a_{ij}$ , are zero, then

det A = a_{ij} A_{ij} .

Theorem 12. The sum of the products of the entries of one row (column) with the cofactors of the corresponding entries of a different row (column) equals zero:

a_{i 1} A_{k 1} + a_{i 2} A_{k 2} + \dots + a_{in} A_{k n} = 0 (i \neq = k) .

(Theorems 11 and 12 together are equivalent to the identity $A \cdot \tilde{A} = det A \cdot E$ , which underlies the formula for the inverse matrix.)

Methods for Computing Determinants

For matrices of order higher than $3$ , direct formulas become unwieldy, so two techniques are used instead.

I. Order-Reduction Method

Using elementary transformations (adding to one row another row multiplied by a number — this does not change the determinant), zero out all entries of some column except one, then expand along that column.

Example. In the determinant

102 - 2 43 3 - 1 1

zero out the first column by subtracting twice the first row from the third ( $R_{3} \to R_{3} - 2 R_{1}$ ):

= 100 - 2 47 3 - 1 - 5 = 1 \cdot A_{11} = 1 \cdot 47 - 1 - 5 = 1 \cdot (4 \cdot (- 5) - (- 1) \cdot 7) = - 13.

II. Reduction to Triangular Form

The determinant is reduced by elementary transformations to triangular form; it then equals the product of the diagonal entries (accounting for the sign from row swaps and factored-out multipliers).

Example:

21 - 3 052 - 3 11 = R_{1} \leftrightarrow R_{2} - 12 - 3 502 1 - 3 1 = R_{2} - 2 R_{1} R_{3} + 3 R_{1} - 100 5 - 10 17 1 - 5 4 .

Next, $R_{3} \to R_{3} + \frac{17}{10} R_{2}$ gives $a_{33} = 4 - \frac{17}{2} = - \frac{9}{2}$ :

= - 100 5 - 10 0 1 - 5 - \frac{9}{2} = - (1 \cdot (- 10) \cdot (- \frac{9}{2})) = - 45.

Properties of Determinants

Determinants have a number of properties (the determinant of the identity and triangular matrix, behavior under transposition, row swaps, scalar multiplication, $det (A B) = det A \cdot det B$ , and others), which are systematically worked out in a separate article, Properties of Determinants.

Related works