MTH2021 · Linear Algebra with Applications
Determinants
Week 3 introduces the determinant through recursive cofactor (Laplace) expansion, the effect of row operations on its value, and the headline properties det(AB) = det(A)det(B), det(Aᵀ) = det(A) and det(A⁻¹) = 1/det(A). The determinant gives a one-number invertibility test (A invertible ⇔ det(A) ≠ 0) and drives Cramer's rule. Determinants are examined in Quiz 3 and in Test 2 (Weeks 3–7), and the multiplicative property is a favourite short-proof target in the final.
What this chapter covers
- 01Recursive definition: det[a] = a, det of a 2×2 = ad − bc; minor Mᵢⱼ and cofactor Cᵢⱼ = (−1)^{i+j}Mᵢⱼ
- 02Cofactor (Laplace) expansion along ANY row or column gives the same value
- 03A row or column of zeros ⇒ det = 0; triangular matrix ⇒ det = product of diagonal entries
- 04Row-operation effects: scale a row by k ⇒ det × k; swap two rows ⇒ det × (−1); add a multiple of one row to another ⇒ det unchanged
- 05det(kA) = kⁿ det(A); det(EA) = det(E)det(A); det(AB) = det(A)det(B); det(A⁻¹) = 1/det(A)
- 06det(Aᵀ) = det(A) — expansion along a row equals expansion along the matching column
- 07Invertibility test: A invertible ⇔ det(A) ≠ 0 (extends the Equivalent-Statements list)
- 08Applications: computing det via reduction to echelon form; Cramer's rule
Determinant by cofactor expansion, checked two ways
- +1Expand along row 1: det(A) = 1·C₁₁ + 2·C₁₂ + 3·C₁₃, with cofactors carrying signs (+ − +).
- +1Compute the three 2×2 minors: M₁₁ = det[[1,0],[1,1]] = 1; M₁₂ = det[[2,0],[4,1]] = 2; M₁₃ = det[[2,1],[4,1]] = 2·1 − 1·4 = −2.
- +1Combine with signs: det(A) = 1·(+1) − 2·(2) + 3·(−2) = 1 − 4 − 6 = −9.
- +1Check by expanding along column 3 (entries 3, 0, 1; signs + − +): 3·det[[2,1],[4,1]] − 0 + 1·det[[1,2],[2,1]] = 3·(2−4) + 1·(1−4) = 3·(−2) + (−3) = −6 − 3 = −9 ✓.
Key terms
- Minor Mᵢⱼ
- The determinant of the (n−1)×(n−1) matrix left after deleting row i and column j of A.
- Cofactor Cᵢⱼ
- The signed minor Cᵢⱼ = (−1)^{i+j} Mᵢⱼ; the sign follows the checkerboard pattern starting with + in the top-left.
- Cofactor (Laplace) expansion
- det(A) = Σⱼ (A)ᵢⱼ Cᵢⱼ along any fixed row i (or column j). The value is independent of the row or column chosen.
- Multiplicative property
- det(AB) = det(A)det(B) for square A, B of the same size; combined with det(I) = 1 it gives det(A⁻¹) = 1/det(A).
- Invertibility test
- A square matrix A is invertible if and only if det(A) ≠ 0; a zero determinant signals a singular matrix with linearly dependent rows/columns.
- Cramer's rule
- For an invertible A, the solution of Ax = b has xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A with b.
Determinants FAQ
Which row or column should I expand along?
Whichever has the most zeros, because each zero entry removes an entire cofactor term. Cofactor expansion gives the same determinant along any row or column (that is a theorem), so you are free to choose the easiest one. For larger matrices it is often faster to row-reduce to triangular form and multiply the diagonal, tracking the det factors from each operation.
How do row operations change the determinant?
Scaling a row by k multiplies det by k; swapping two rows multiplies det by −1; adding a multiple of one row to another leaves det unchanged. This is why reducing to echelon form and multiplying the diagonal entries works — you just accumulate the k's and sign flips you introduced along the way.
Why does det(A) ≠ 0 mean A is invertible?
If A is invertible then det(A)det(A⁻¹) = det(I) = 1, so det(A) cannot be 0. Conversely a zero determinant forces the rows (and columns) to be linearly dependent, so RREF(A) ≠ I and A is singular. This adds det(A) ≠ 0 to the Equivalent-Statements list.
Is det(AB) = det(A)det(B) really always true?
Yes, for square matrices of the same size — it is one of the most-used theorems in the unit and a common handwritten-exam proof (via elementary matrices and det(EA) = det(E)det(A)). Note there is no such rule for sums: det(A + B) is generally not det(A) + det(B).
Where do determinants get tested?
Quiz 3 covers Week-3 material, and determinants fall inside Test 2 (Weeks 3–7). In the final you may be asked to compute a determinant efficiently and to prove a property such as det(AB) = det(A)det(B) or the row-swap sign change, so practise both.
Exam move
Get the cofactor signs (−1)^{i+j} automatic and always expand along the row or column with the most zeros; for anything 4×4 or larger, reduce to triangular form and track the row-operation factors instead. Learn the property list cold — det(AB) = det(A)det(B), det(Aᵀ) = det(A), det(A⁻¹) = 1/det(A), det(kA) = kⁿdet(A) — because the exam tests them as short proofs. Determinants are examined in Quiz 3 and Test 2 (Weeks 3–7). When an expansion goes wrong, ask Sia to check your minors and signs line by line.
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