Source: Abstract Linear Algebra, UIUC
Tags: determinant, cofactor expansion, Laplace expansion, triangular matrix, Gaussian elimination, row swap, det(AB), matrix determinant properties, 2x2 determinant, 3x3 determinant, linear algebra
Difficulty: Intermediate Prerequisites: Matrix rank and inverse (Part 1 notes), comfort with elementary row operations and matrix notation.
The determinant is a single number that encodes a surprising amount of information about a square matrix. It tells you whether the matrix is invertible, how it scales volumes under the associated linear transformation, and it appears in formulas for eigenvalues, cross products, and change-of-variable integrals. In this course, you need to know how to compute determinants efficiently and how the key properties connect to rank and invertibility. If you missed the rank and inverse material, review that first, because determinant properties lean on it heavily.
A determinant is a scalar value computed from a square matrix. For 2 × 2 it is ad – bc; for larger matrices you use cofactor expansion or row-reduce to triangular form and multiply the diagonal. A zero determinant means the matrix is singular; a nonzero determinant means it is invertible.
Determinant (det)
A scalar-valued function of a square matrix that is zero precisely when the matrix is singular. In simple terms, it is a single number that tells you whether the matrix "collapses" space (det = 0) or preserves dimensionality (det ≠ 0).
Minor (Mᵢⱼ)
The determinant of the (n–1) × (n–1) submatrix obtained by deleting row i and column j. Think of it as the determinant of "everything left over" when you remove one row and one column.
Cofactor (Cᵢⱼ)
The signed minor: Cᵢⱼ = (–1)^(i+j) × Mᵢⱼ. The sign alternates in a checkerboard pattern starting with + in the top-left corner.
Cofactor expansion (Laplace expansion)
A method for computing det(A) by expanding along any single row or column: sum each entry times its cofactor. In simple terms, you break a big determinant into smaller ones, one row or column at a time.
Triangular matrix
A square matrix where all entries above the diagonal (lower triangular) or below the diagonal (upper triangular) are zero. Its determinant is simply the product of the diagonal entries, which makes computation very fast.
1 × 1 matrix: det([a]) = a. The determinant is just the entry itself.
2 × 2 matrix: For A = [[a₁₁, a₁₂], [a₂₁, a₂₂]], det(A) = a₁₁a₂₂ – a₁₂a₂₁. Memorise this; it is used constantly inside larger cofactor expansions.
Choose any row or column. For each entry in that row or column, multiply the entry by its cofactor and sum the results.
Expanding along a row or column with more zeros means fewer terms to compute. Always scan for the row or column with the most zeros before starting.
The result is the same regardless of which row or column you choose.
Triangular matrices: det = product of diagonal entries. This is the fastest route when the matrix is already triangular or close to it.
Row/column swap: swapping two rows (or two columns) multiplies the determinant by –1.
Repeated rows/columns: if two rows (or two columns) are identical, det = 0.
Scalar row multiplication: multiplying one row by scalar k multiplies det by k.
Row addition: adding a multiple of one row to another row does not change the determinant.
Product rule: det(AB) = det(A) · det(B). This holds for any two square matrices of the same size.
Transpose: det(Aᵀ) = det(A).
Inverse: det(A⁻¹) = 1 / det(A), provided A is invertible.
Row-reduce the matrix to upper triangular form, keeping track of any row swaps (each swap flips the sign) and any row scalings (each scaling by k multiplies det by k).
Once in triangular form, multiply the diagonal entries together, then apply the accumulated sign and scaling corrections.
This approach is generally faster than cofactor expansion for matrices 4 × 4 and larger.
2 × 2: use the formula directly.
3 × 3: cofactor expansion is quick, especially if you spot a row or column with zeros.
4 × 4 and above: row-reduce to triangular form unless the matrix already has a very favourable structure for expansion.
2 × 2 determinant: det(A) = a₁₁a₂₂ – a₁₂a₂₁
Cofactor expansion along row i: det(A) = Σⱼ aᵢⱼ · (–1)^(i+j) · Mᵢⱼ
Product rule: det(AB) = det(A) · det(B)
Triangular matrix determinant: det(A) = a₁₁ · a₂₂ · … · aₙₙ
Determinants appear in the change-of-variables formula for multivariable integrals (the Jacobian determinant). In physics, the cross product of two 3D vectors is computed via a 3 × 3 determinant. In computer graphics, determinants help test whether a set of points is oriented clockwise or anticlockwise.
Students often forget that each row swap flips the sign of the determinant. Losing track of an odd number of swaps during Gaussian elimination gives the wrong sign.
A common mistake is thinking that adding a multiple of one row to another changes the determinant. It does not.
Some students try to compute cofactor expansion for large matrices without first simplifying. Row-reduce to create zeros first, then expand.
Confusing minors and cofactors: the cofactor includes the (–1)^(i+j) sign; the minor does not.
⚠️ "Evaluate the determinant" is a staple exam question. You will be expected to choose an efficient method and show your working cleanly.
⚠️ Properties questions (true/false, multiple choice) test whether you know how row operations affect the determinant. Memorise the effect of each operation.
⚠️ The product rule det(AB) = det(A) · det(B) often appears in proof-style or short-answer questions. Know it cold.
⚠️ det(A) = 0 ⟺ A is singular ⟺ rank(A) < n. This equivalence links determinants to everything in Part 1.
True or false: Swapping two rows of a matrix does not affect the determinant.
Fill in the blank: The determinant of a triangular matrix equals the ______ of its diagonal entries.
True or false: det(AB) = det(A) + det(B).
Fill in the blank: If two columns of a matrix are identical, det = ______.
True or false: Multiplying a single row by 5 multiplies the determinant by 5.
Answers: 1. False (it negates the determinant). 2. Product. 3. False (det(AB) = det(A) · det(B)). 4. 0. 5. True.
Q: You are given a 4 × 4 matrix. Describe the most efficient general strategy for computing its determinant by hand.
A: Row-reduce to upper triangular form using row replacement operations (which do not change the determinant), tracking any row swaps. Then multiply the diagonal entries and apply the sign correction for an odd or even number of swaps.
Q: Explain why a matrix with two identical rows has determinant zero.
A: Swapping those two identical rows leaves the matrix unchanged, but the swap rule says the determinant must change sign. The only number equal to its own negation is zero, so det = 0.
Q: If det(A) = 3 and det(B) = –2, what is det(AB)?
A: det(AB) = det(A) · det(B) = 3 × (–2) = –6.
Q: For a 3 × 3 upper triangular matrix with diagonal entries 2, –1, 4, what is the determinant?
A: det = 2 × (–1) × 4 = –8.
Determinants tie back to rank and invertibility from Part 1: det ≠ 0 is equivalent to full rank. Looking ahead, determinants are central to the characteristic polynomial det(A – λI) = 0, which defines eigenvalues. They also appear in Cramer's rule for solving systems of equations and in the formula for the cross product in ℝ³.
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