Source: Friedberg, Insel, Spence – Linear Algebra, 4th Ed.
Tags: determinant properties, multiplicative property, det(AB), transpose determinant, Cramer's rule, classical adjoint, invertibility, elementary matrices, n-linear function, alternating function, parallelepiped volume, orientation, linear algebra
Difficulty: Intermediate–Advanced | Prerequisites: Sections 4.1–4.2, elementary matrices (Section 3.1), rank (Chapter 3).
Sections 4.1 and 4.2 defined determinants and showed how to compute them. This final stretch of Chapter 4 shifts the focus to what determinants can do. The headline results are: determinants multiply (det(AB) = det(A)·det(B)), a matrix is invertible exactly when its determinant is nonzero, and the determinant of a transpose equals the determinant of the original. Cramer's rule gives a formula for solving n×n systems, though it is more useful theoretically than computationally. Section 4.5 (optional) rounds things off by characterising the determinant uniquely through three properties: n-linearity, alternating behaviour, and det(I) = 1.
The determinant is a multiplicative function: det(AB) = det(A)·det(B). A matrix is invertible if and only if its determinant is nonzero. Transposing a matrix does not change its determinant, so every row-based determinant result has a column-based twin. Cramer's rule solves Ax = b by ratios of determinants but is computationally expensive.
Multiplicative property (Theorem 4.7)
det(AB) = det(A) · det(B) for any n×n matrices A and B. In simple terms, the determinant of a product is the product of the determinants. This is one of the most powerful facts in the chapter.
Transpose determinant (Theorem 4.8)
det(Aᵗ) = det(A). In simple terms, swapping rows and columns does not affect the determinant. This means any theorem about rows of a determinant has an automatic column analogue.
Cramer's rule (Theorem 4.9)
For a system Ax = b with det(A) ≠ 0, the kth unknown is xₖ = det(Mₖ) / det(A), where Mₖ is A with its kth column replaced by b. Think of it as: each unknown is a ratio of two determinants. Elegant, but slow for large systems.
Classical adjoint
The transpose of the cofactor matrix. For n×n matrix A, if C is the matrix whose (i, j) entry is the cofactor cⱼᵢ (note the transposition), then AC = det(A)·I. When A is invertible, A⁻¹ = (1/det(A))·C. In simple terms, it is a way to write the inverse purely in terms of cofactors and the determinant.
n-linear function (Section 4.5)
A function δ: Mₙₓₙ(F) → F that is linear in each row when the remaining n − 1 rows are held fixed. The determinant is the most important example.
Alternating function (Section 4.5)
An n-linear function δ such that δ(A) = 0 whenever two adjacent rows of A are identical. This forces sign changes on row swaps and zeros on matrices with any two equal rows.
Parallelepiped volume
For A ∈ Mₙₓₙ(R) with rows a₁, …, aₙ, the quantity |det(A)| equals the n-dimensional volume of the parallelepiped with edges a₁, …, aₙ. Generalises the 2D parallelogram-area result from Section 4.1.
Type 1 (row swap): det(E₁) = −1.
Type 2 (scale row by k): det(E₂) = k.
Type 3 (add multiple of one row to another): det(E₃) = 1.
These follow from applying the row-operation rules to the identity matrix, which has determinant 1.
For any A, B ∈ Mₙₓₙ(F): det(AB) = det(A) · det(B).
Proof sketch: handle the case where A is an elementary matrix (use the row-operation rules), handle the case where rank(A) < n (both sides are 0), then write a full-rank A as a product of elementary matrices and apply the elementary case repeatedly.
This is one of the most frequently used properties in later chapters.
A ∈ Mₙₓₙ(F) is invertible ⟺ det(A) ≠ 0.
If A is invertible: det(A⁻¹) = 1 / det(A).
Proof of the forward direction: det(A) · det(A⁻¹) = det(AA⁻¹) = det(I) = 1, so neither factor can be zero.
det(Aᵗ) = det(A) for any A ∈ Mₙₓₙ(F).
Consequence 1: the determinant can be evaluated by cofactor expansion along any column, not just any row.
Consequence 2: elementary column operations affect the determinant the same way their row counterparts do.
Given Ax = b with det(A) ≠ 0, the system has a unique solution.
Each component: xₖ = det(Mₖ) / det(A), where Mₖ is A with column k replaced by b.
Useful when you need a formula for the solution (e.g., proving existence of integer solutions when det(A) = ±1 and coefficients are integers).
Not useful for computation: it requires evaluating n + 1 determinants of n×n matrices, far more work than Gaussian elimination.
System: x₁ + 2x₂ + 3x₃ = 2, x₁ + x₃ = 3, x₂ + x₂ − x₃ = 1.
A = [[1,2,3],[1,0,1],[1,1,−1]], b = (2,3,1)ᵗ. det(A) = 6.
x₁ = det([[2,2,3],[3,0,1],[1,1,−1]]) / 6 = 15/6 = 5/2.
x₂ = det([[1,2,3],[1,3,1],[1,1,−1]]) / 6 = −6/6 = −1.
x₃ = det([[1,2,2],[1,0,3],[1,1,1]]) / 6 = 3/6 = 1/2.
|det(A)| = n-dimensional volume of the parallelepiped spanned by the row vectors.
Example: rows (1,−2,1), (1,0,−1), (1,1,1) give |det| = 6, which is the volume of the resulting rectangular parallelepiped (side lengths √6, √2, √3, and √6·√2·√3 = 6).
The sign of the determinant encodes orientation: det(Q) > 0 means the ordered basis induces a right-handed coordinate system.
The determinant is the unique function δ: Mₙₓₙ(F) → F that is:
n-linear (linear in each row when the others are held fixed),
alternating (δ(A) = 0 when two adjacent rows are identical),
normalised: δ(I) = 1.
Alternating functions satisfy δ(B) = −δ(A) when B is obtained from A by swapping any two rows, and δ(A) = 0 when A has any two identical rows (not just adjacent ones).
Any alternating n-linear function equals k · det for some scalar k. The normalisation δ(I) = 1 forces k = 1.
Multiplicative property:
det(AB) = det(A) · det(B)
Inverse determinant:
det(A⁻¹) = 1 / det(A)
Transpose determinant:
det(Aᵗ) = det(A)
Cramer's rule:
xₖ = det(Mₖ) / det(A), where Mₖ = A with column k replaced by b.
Scaling rule:
det(kA) = kⁿ · det(A), for an n×n matrix A.
Similar matrices:
If A and B are similar (B = P⁻¹AP), then det(A) = det(B).
Swapping two rows (or columns) flips the sign.
Scaling a row (or column) by k multiplies det by k.
Adding a multiple of one row to another leaves det unchanged.
The determinant of an upper triangular matrix is the product of its diagonal entries. In particular, det(I) = 1.
Two identical rows (or columns) means det = 0.
det(AB) = det(A) · det(B).
A is invertible ⟺ det(A) ≠ 0. If invertible, det(A⁻¹) = 1/det(A).
det(Aᵗ) = det(A).
Similar matrices have the same determinant.
The multiplicative property det(AB) = det(A)·det(B) underpins results in group theory (the determinant is a group homomorphism from GL(n, F) to F*). In physics and differential geometry, the determinant of the Jacobian matrix measures how volumes transform under changes of variables, a fact used constantly in multivariable calculus (change-of-variables theorem). Cramer's rule, while computationally expensive, guarantees integer solutions to integer-coefficient systems when |det(A)| = 1, which matters in combinatorics and integer programming.
Students sometimes write det(A + B) = det(A) + det(B). This is false. The multiplicative rule applies to products, not sums.
det(Aᵗ) = det(A), not −det(A). The transpose does not flip the sign.
Cramer's rule replaces a column of A with b, not a row. If the question says "replace row k with bᵗ," that is a different (and incorrect) formulation.
det(kA) = kⁿ · det(A), not k · det(A). Scaling the entire matrix scales each of the n rows by k, so the factor is kⁿ.
⚠️ det(AB) = det(A) · det(B) is used in many proofs and is a near-certain exam topic, both as a computation tool and in true/false questions.
⚠️ The equivalence "A invertible ⟺ det(A) ≠ 0" ties together Chapters 3 and 4 and feeds into Chapter 5. Expect it to appear in multiple forms.
⚠️ det(Aᵗ) = det(A) justifies expanding along columns. True/false trap: "det(Aᵗ) = −det(A)" is false.
⚠️ Cramer's rule is commonly tested via small worked examples (2×2 or 3×3 systems). Know the formula, know when it applies (det ≠ 0), and know why it is impractical for large n.
⚠️ The formula det(kA) = kⁿ det(A) comes up in questions like "under what conditions is det(−A) = det(A)?" (Answer: when n is even.)
⚠️ For Section 4.5 (if covered): the axiomatic characterisation is the conceptual capstone. Knowing the three properties (n-linear, alternating, δ(I) = 1) and that they determine det uniquely is the key takeaway.
True or false: For any A, B ∈ Mₙₓₙ(F), det(AB) = det(A) + det(B).
If det(A) = 5, what is det(A⁻¹)?
True or false: det(Aᵗ) = −det(A).
Fill in the blank: det(3A) = ___ · det(A) when A is 4×4.
True or false: If A and B are similar, then det(A) = det(B).
(Answers: 1. False, det(AB) = det(A)·det(B). 2. 1/5. 3. False, det(Aᵗ) = det(A). 4. 3⁴ = 81. 5. True.)
Q: Prove that if A is invertible, then det(A) ≠ 0.
A: If A is invertible, then AA⁻¹ = I. By the multiplicative property, det(A) · det(A⁻¹) = det(I) = 1. Since 1 ≠ 0, neither det(A) nor det(A⁻¹) can be zero.
Q: Use Cramer's rule to solve: 2x₁ + x₂ − 3x₃ = 5, x₁ − 2x₂ + x₃ = 10, 3x₁ + 4x₂ − 2x₃ = 0.
A: Compute det(A), then det(M₁), det(M₂), det(M₃) by replacing the appropriate column with (5, 10, 0)ᵗ. Divide each by det(A). (Work through the 3×3 determinant computations using your preferred method.)
Q: Let A ∈ M₃ₓ₃(F). What is det(−A) in terms of det(A)?
A: det(−A) = (−1)³ · det(A) = −det(A). In general, det(−A) = (−1)ⁿ · det(A), so for odd n the sign flips and for even n it stays.
Q: True or false – if E is an elementary matrix, then det(E) = ±1.
A: False. A type 2 elementary matrix (scaling a row by k ≠ ±1) has det(E) = k, which need not be ±1.
Q: State the three properties that uniquely characterise the determinant.
A: The determinant is the unique function δ: Mₙₓₙ(F) → F such that (1) δ is n-linear, (2) δ is alternating (δ(A) = 0 when two adjacent rows are identical), and (3) δ(I) = 1.
Q: The volume of the parallelepiped with edges (1, −2, 1), (1, 0, −1), (1, 1, 1) is ___.
A: |det([[1,−2,1],[1,0,−1],[1,1,1]])| = |6| = 6.
The multiplicative property and the invertibility criterion connect determinants directly to the theory of linear transformations and change-of-basis matrices from Chapter 2. In Chapter 5, the characteristic polynomial det(A − λI) uses the determinant to find eigenvalues, and the fact that similar matrices share the same determinant (Property 9) ensures eigenvalues are well-defined for linear operators, not just matrices. In more advanced courses, the determinant generalises to the exterior algebra and differential forms, where "alternating multilinear" becomes the defining concept.
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