Determinants: Axiomatic Definition, Laplace Expansion and Properties

The determinant of an n × n matrix is the unique number det A that is linear in each row, changes sign when two rows are swapped, and equals 1 for the identity matrix; it can be computed by Laplace (cofactor) expansion, by row reduction, or for 2 × 2 matrices by ad − bc. In brief: det [[3, 1], [2, 5]] = 3 · 5 − 1 · 2 = 13, and a square matrix is invertible exactly when its determinant is nonzero. The absolute value of the determinant is the area (or volume) scaling factor of the corresponding linear map.

The determinant condenses a square matrix into one number that detects invertibility, measures volume and appears in Cramer’s rule and in eigenvalue theory. This guide gives the axiomatic definition, the formulas for 2 × 2 and 3 × 3 matrices, the Laplace expansion, the effect of row operations, the main properties such as det(AB) = det A · det B, and the geometric meaning, with worked examples and a table.

How is the determinant defined?

Viewing a matrix as a list of rows, the determinant is the unique function det with three properties:

• Linearity in each row: if one row is a sum or a multiple, the determinant splits or scales accordingly, with the other rows unchanged.
• Alternating: swapping two rows changes the sign; consequently a matrix with two equal rows has the determinant 0.
• Normalization: det I = 1.

These axioms determine det completely. The explicit Leibniz formula is det A = Σ sgn(σ) a_1σ(1) a_2σ(2) … a_nσ(n) over all permutations σ, where sgn σ = ±1 depends on the parity of σ. It has n! terms and is rarely used for computation.

How do you compute 2 × 2 and 3 × 3 determinants?

For 2 × 2: det [[a, b], [c, d]] = ad − bc. For 3 × 3, the rule of Sarrus: add the products along the three diagonals going down to the right and subtract the products along the three going down to the left. The rule of Sarrus works only for 3 × 3.

Example 1: a 2 × 2 determinant

det [[3, 1], [2, 5]] = 15 − 2 = 13. And det [[1, 2], [2, 4]] = 4 − 4 = 0, because the second row is twice the first.

Example 2: a 3 × 3 determinant by Sarrus

A = [[2, 0, 1], [1, 3, 2], [1, 1, 4]]. Positive terms: 2 · 3 · 4 + 0 · 2 · 1 + 1 · 1 · 1 = 25. Negative terms: 1 · 3 · 1 + 2 · 2 · 1 + 0 · 1 · 4 = 7. det A = 18.

What is the Laplace expansion?

The cofactor of the entry a_ij is C_ij = (−1)^i+j M_ij, where the minor M_ij is the determinant of the matrix with row i and column j deleted. The expansion along row i is det A = a_i1C_i1 + a_i2C_i2 + … + a_inC_in, and the same holds for any column. The signs form a checkerboard pattern starting with + in the corner. Expanding along a row or column with many zeros saves work.

Example 3: the same determinant, expanded along row 2

For A above, row 2 is (1, 3, 2). C₂₁ = −det [[0, 1], [1, 4]] = −(0 − 1) = 1. C₂₂ = +det [[2, 1], [1, 4]] = 8 − 1 = 7. C₂₃ = −det [[2, 0], [1, 1]] = −2. Then det A = 1 · 1 + 3 · 7 + 2 · (−2) = 1 + 21 − 4 = 18 ✓.

Example 4: a 4 × 4 determinant with zeros

B = [[1, 0, 2, 0], [3, 1, 0, 0], [0, 2, 1, 4], [1, 0, 0, 2]]. Expand along row 1: det B = 1 · det [[1, 0, 0], [2, 1, 4], [0, 0, 2]] + 2 · det [[3, 1, 0], [0, 2, 4], [1, 0, 2]] (the sign for a₁₃ is +). The first 3 × 3 determinant equals 1 · (2 − 0) − 0 + 0 = 2. The second: 3(4 − 0) − 1(0 − 4) + 0 = 12 + 4 = 16. So det B = 2 + 2 · 16 = 34.

How do row operations compute a determinant?

Row reduction is the efficient method for larger matrices. Swapping rows changes the sign, scaling a row by c multiplies the determinant by c, and adding a multiple of one row to another changes nothing. For a triangular matrix, the determinant is the product of the diagonal entries.

Example 5: by elimination

M = [[1, 2, 3], [4, 5, 6], [7, 8, 10]]. R₂ − 4R₁ = (0, −3, −6), R₃ − 7R₁ = (0, −6, −11), then R₃ − 2R₂ = (0, 0, 1). The triangular matrix has the diagonal 1, −3, 1, so det M = −3. Check with Sarrus: 50 + 84 + 96 − (105 + 48 + 80) = 230 − 233 = −3 ✓.

Example 6: a triangular matrix

[[1, 2, 3, 4], [0, 2, 5, 6], [0, 0, 3, 7], [0, 0, 0, 4]] has the determinant 1 · 2 · 3 · 4 = 24.

Example 7: the Vandermonde determinant

det [[1, 1, 1], [a, b, c], [a², b², c²]] = (b − a)(c − a)(c − b). For (a, b, c) = (1, 2, 3): 1 · 2 · 1 = 2. Direct check on [[1, 1, 1], [1, 2, 3], [1, 4, 9]]: 1(18 − 12) − 1(9 − 3) + 1(4 − 2) = 6 − 6 + 2 = 2 ✓.

What are the main properties?

• det(AB) = det A · det B.
• det A^T = det A, so statements about rows hold for columns.
• det(kA) = kⁿ det A for an n × n matrix.
• A is invertible if and only if det A ≠ 0, and then det A⁻¹ = 1/det A.
• The determinant is zero if a row is a combination of the other rows, in particular if two rows are equal or a row is zero.
• Similar matrices have equal determinants, and det A is the product of the eigenvalues.

Example 8: the product rule

A = [[1, 2], [3, 4]] has det A = −2; B = [[0, 1], [1, 0]] has det B = −1. AB = [[2, 1], [4, 3]] has the determinant 6 − 4 = 2 = (−2)(−1) ✓.

Example 9: scaling a matrix

3A = [[3, 6], [9, 12]] has det = 36 − 54 = −18 = 3² · (−2) ✓. Note that the factor is kⁿ and not k.

Example 10: a parameter

For which k is [[1, k, 0], [k, 1, k], [0, k, 1]] singular? det = 1(1 − k²) − k(k − 0) + 0 = 1 − 2k². It is 0 for k = ±1/√2.

What does the determinant mean geometrically?

The absolute value of det [[a, b], [c, d]] is the area of the parallelogram spanned by the rows (a, b) and (c, d), and for 3 × 3 matrices it is the volume of the parallelepiped spanned by the rows. The sign shows whether orientation is preserved. A linear map multiplies all areas (volumes) by |det|.

Example 11: an area and a volume

The vectors (3, 1) and (1, 2) span a parallelogram of area |6 − 1| = 5. The rows (1, 0, 0), (1, 2, 0), (1, 1, 3) span a parallelepiped of volume |det| = 1 · (6 − 0) = 6.

Table of determinant rules

Where do determinants come from?

Determinants appeared before matrices: the Japanese mathematician Seki Takakazu used them around 1683 to solve systems of equations, and Leibniz described a similar idea in 1693. Gabriel Cramer published his rule in 1750, Pierre-Simon Laplace gave the expansion by rows and columns in 1772, and Augustin-Louis Cauchy proved the product theorem and used the word “determinant” in its modern sense in 1812 (Gauss had used the term earlier for a different quantity, the discriminant of a quadratic form). The axiomatic characterization is a twentieth-century exposition.

Which mistakes are common?

Using Sarrus for larger matrices

The diagonal rule works only for 3 × 3. For larger matrices use expansion or elimination.

Wrong checkerboard signs

The cofactor sign is (−1)^i+j. Forgetting it is the most frequent error in an expansion.

det(A + B) = det A + det B

This is false. The determinant is multiplicative, not additive: det(A + B) ≠ det A + det B in general.

Forgetting the effect of row operations

When scaling a row by c, divide the final result by c. Swapping rows flips the sign.