Difficulty: Intermediate | Prerequisites: Matrix notation, basic arithmetic with fractions
The determinant is a single number that encodes whether a square matrix is invertible, how it scales area or volume, and how it behaves under row operations. Nearly every topic in this chapter (inverses, Cramer's rule, eigenvalues later) depends on being able to compute determinants quickly and accurately. This material assumes you can multiply and add fractions without trouble.
The determinant of a square matrix is a single scalar that tells you whether the matrix is invertible (non-zero determinant) or singular (zero determinant). For 2×2 matrices, it is ad – bc. For 3×3, you expand along a row or column using cofactors. Several properties let you simplify the computation, especially for triangular matrices where the determinant is just the product of the diagonal entries.
Determinant (det(A) or |A|)
A scalar value computed from a square matrix. For a 2×2 matrix [a, b; c, d], det = ad – bc. Think of it as a measure of how the matrix scales area (2D) or volume (3D).
Minor (Mᵢⱼ)
The determinant of the submatrix formed by deleting row i and column j from A. In simple terms, you cross out one row and one column, then take the determinant of what is left.
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 the determinant by expanding along any row or column. You multiply each entry by its cofactor and sum the results. In simple terms, you pick a row (or column), and each entry in that row "votes" for the determinant weighted by its cofactor.
Triangular matrix
A square matrix where all entries above the diagonal (lower triangular) or below the diagonal (upper triangular) are zero. Its determinant equals the product of its diagonal entries.
Row echelon form
A matrix where each leading entry is to the right of the one above it, and all entries below each leading entry are zero. Useful for computing determinants because the result is a triangular matrix.
Singular matrix
A square matrix with det(A) = 0. It has no inverse, and the associated linear system either has no solution or infinitely many solutions.
For A = [a, b; c, d]: det(A) = ad – bc. Straightforward, and worth memorising as a reflex.
Expand along any row or column. The standard choice is the first row:
det(A) = a₁₁ · C₁₁ + a₁₂ · C₁₂ + a₁₃ · C₁₃
where each cofactor C₁ⱼ = (−1)^(1+j) · (the 2×2 determinant you get after deleting row 1 and column j).
Tip: expand along the row or column with the most zeros. Every zero entry kills its term entirely, saving you computation.
Worked example (9.2.1a)
A = [3, 0, 4; 2, 3, 2; 0, 5, −1]. Expand along row 1:
det(A) = 3 · det[3, 2; 5, −1] – 0 · (...) + 4 · det[2, 3; 0, 5]
= 3(−13) – 0 + 4(10) = −39 + 40 = 1
The zero in position (1,2) eliminated one full 2×2 determinant calculation.
Worked example (9.2.1b)
B = [2, −4, 3; 3, 1, 2; 1, 4, −1]. Expand along row 1:
det(B) = 2 · det[1, 2; 4, −1] – (−4) · det[3, 2; 1, −1] + 3 · det[3, 1; 1, 4]
= 2(−9) + 4(−5) + 3(11) = −18 – 20 + 33 = −5
Note the sign pattern: +, –, + for positions (1,1), (1,2), (1,3). The entry −4 in position (1,2) gets multiplied by −1, which flips it to +4.
More results from the homework
det(C) = −23 for C = [2, 3, −4; 4, 0, 2; 5, 1, 6] (9.2.1c)
det(D) = 4 for D = [4, 3, 0; 6, 5, 2; 4, 7, 3] (9.2.1d)
These properties let you simplify before computing:
Multiplying a row by scalar k multiplies det by k. So if B is obtained from A by multiplying row 3 by 3, then det(B) = 3 · det(A). (Used in 9.2.3: det(B) = 3 · 7 = 21.)
Swapping two rows flips the sign of det. Each swap multiplies det by −1.
Adding a multiple of one row to another does not change det.
For larger matrices, row-reduce to triangular form, then take the product of the diagonal entries. Adjust for any row swaps (each swap flips the sign).
Worked example (9.3.4)
A is a 4×4 matrix. After row reduction (with two row swaps), the diagonal of the upper triangular form is (−3, −4, 35/12, 0). Because one diagonal entry is 0, the determinant is 0.
The factor (−1)² from two row swaps would have given +1, but it does not matter: a zero on the diagonal means det(A) = 0 regardless.
det(AB) = det(A) · det(B) (Chapter 9 Review, Problem 6)
Even though AB may not equal BA, their determinants are always equal: det(AB) = det(A) det(B) = det(B) det(A) = det(BA). Determinants are scalars, and scalar multiplication is commutative.
Orthogonal matrix proof (Chapter 9 Review, Problem 7)
If AᵀA = I, then det(AᵀA) = det(I) = 1. Since det(Aᵀ) = det(A), we get det(A)² = 1, so det(A) = ±1.
M(a) = [a, eᵃ, 1–a; 0, a²−1, 7a; 0, 0, 2–a]. This is upper triangular, so det(M) = a(a² – 1)(2 – a).
M is invertible when det(M) ≠ 0. Setting each factor to zero: a = 0, a² – 1 = 0 (a = ±1), 2 – a = 0 (a = 2). So M is invertible for all a not in {−1, 0, 1, 2}.
Given three matrices L, M, N with the same entries a through f rearranged, and told that det(M) = 10 and det(N) = −5:
From det(M) = 10, deduce af – eb = 10.
From det(N) = −5, deduce ad – cb = 5.
Expand det(L) using cofactor expansion to get 5(af – be) – 3(ad – bc) = 5(10) – 3(5) = 35.
This type of problem tests whether you can express a new determinant in terms of known sub-expressions rather than computing from scratch.
Determinants appear in physics (calculating the cross product of two vectors is a 3×3 determinant), in calculus (the Jacobian determinant for change of variables in multiple integrals), and in economics (testing whether a system of supply and demand equations has a unique equilibrium).
Students often expand along the first row out of habit even when another row or column has more zeros. Expanding along a row or column with zeros saves significant time.
The sign pattern in cofactor expansion trips people up. Remember: position (i,j) carries the sign (−1)^(i+j). The top-left is always positive.
Students sometimes think det(A + B) = det(A) + det(B). This is false. The determinant is not a linear function of the matrix as a whole.
A row swap changes the sign of the determinant. Students who row-reduce to triangular form sometimes forget to account for the swaps, getting the correct absolute value but the wrong sign.
⚠️ Cofactor expansion of a 3×3 matrix is almost guaranteed on the exam. Practise until the sign pattern is automatic.
⚠️ Problems that give det(M) and det(N) and ask for det(L) in terms of them test whether you can factorise a new expansion in terms of known quantities. Do not try to compute from scratch.
⚠️ Know the three row-operation rules cold: scaling a row scales det, swapping rows flips the sign, adding a multiple of one row to another leaves det unchanged.
⚠️ "For what values of a is M(a) invertible?" means: find where det(M(a)) = 0, then exclude those values.
True or false: det(A + B) = det(A) + det(B).
False.
True or false: Swapping two rows of a matrix multiplies the determinant by −1.
True.
Fill in the blank: The determinant of a triangular matrix equals ___.
The product of its diagonal entries.
True or false: det(AB) = det(A) · det(B).
True.
Fill in the blank: If AᵀA = I, then det(A) = ___.
±1.
Q: Compute the determinant of A = [3, 0, 4; 2, 3, 2; 0, 5, −1] by expanding along the first row.
A: det(A) = 3(−13) – 0 + 4(10) = −39 + 40 = 1.
Q: Matrix B is obtained from A by multiplying row 3 of A by 3. If det(A) = 7, what is det(B)?
A: det(B) = 3 · det(A) = 21.
Q: A 4×4 matrix is row-reduced to upper triangular form with diagonal entries (−3, −4, 35/12, 0) after two row swaps. What is det(A)?
A: The product of diagonal entries is 0 (because one entry is 0), so det(A) = 0 regardless of the row swaps.
Q: Prove that |AB| = |BA| for square matrices A and B.
A: det(AB) = det(A) det(B). Since determinants are scalars, det(A) det(B) = det(B) det(A) = det(BA). Matrix multiplication is not commutative, but multiplication of real numbers is.
Q: For which values of a is M(a) = [a, eᵃ, 1–a; 0, a²−1, 7a; 0, 0, 2–a] invertible?
A: det(M) = a(a² – 1)(2 – a) = a(a−1)(a+1)(2–a). This is zero when a ∈ {−1, 0, 1, 2}. M is invertible for all other values of a.
This connects to matrix inverses because the inverse formula A⁻¹ = (1/det(A)) · adj(A) requires the determinant, and a zero determinant means no inverse exists.
This connects to Cramer's rule because each variable in the solution is a ratio of two determinants.
This connects to eigenvalues (a later topic) because eigenvalues are found by solving det(A – λI) = 0.
determinant, det, cofactor expansion, Laplace expansion, minor, cofactor, sign pattern, triangular determinant, row echelon determinant, det(AB) = det(A)det(B), orthogonal matrix determinant, singular matrix, row operations determinant, 3x3 determinant, cofactor sign, upper triangular, lower triangular, parametric determinant