What this chapter is about
Determinants are special numerical values associated with square matrices. After learning matrices in the previous chapter, you now study determinants because they reveal important properties of a matrix — most crucially, whether a system of linear equations has a unique solution. A determinant converts an entire square array of numbers into a single number that carries geometric and algebraic meaning.
In this chapter, you learn how to compute determinants of 2 × 2 and 3 × 3 matrices, understand their properties that simplify calculations, and use determinants to find the area of a triangle, solve systems of linear equations using Cramer's rule, and determine whether a matrix is invertible. The adjoint of a matrix and the formula for the inverse using determinants are also covered here.
By the end, you should be able to evaluate determinants efficiently using row or column operations, apply the properties to simplify complex expressions, and connect determinants to the solvability of linear systems — a skill essential for higher mathematics and applications in physics and economics.
Key ideas
- The determinant of a 2 × 2 matrix A = [[a, b], [c, d]] is written |A| or det(A) and equals ad − bc.
- For a 3 × 3 matrix, the determinant is computed by expansion along any row or column, using minors and cofactors.
- The minor M_ij of an element a_ij is the determinant of the 2 × 2 matrix obtained by deleting the i-th row and j-th column. The cofactor is C_ij = (−1)^(i+j) × M_ij.
- Key properties: interchanging two rows (or columns) changes the sign; if two rows are identical, the determinant is zero; multiplying a row by k multiplies the determinant by k; adding a multiple of one row to another leaves the determinant unchanged.
- A square matrix is singular (non-invertible) if and only if its determinant is zero.
- The adjoint of a matrix A, written adj(A), is the transpose of the matrix of cofactors. The inverse is A⁻¹ = (1/|A|) × adj(A), valid when |A| ≠ 0.
- Cramer's rule gives the solution of a system of n linear equations in n unknowns using ratios of determinants, provided the coefficient determinant is non-zero.
- The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) equals (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
Formulas and facts to remember
- det([[a, b], [c, d]]) = ad − bc — the product of the main diagonal minus the product of the other diagonal.
- For a 3 × 3 matrix [[a₁, b₁, c₁], [a₂, b₂, c₂], [a₃, b₃, c₃]], expanding along the first row: |A| = a₁(b₂c₃ − b₃c₂) − b₁(a₂c₃ − a₃c₂) + c₁(a₂b₃ − a₃b₂).
- |kA| = k^n × |A| for an n × n matrix A.
- |AB| = |A| × |B| for square matrices of the same order.
- |A^T| = |A| — the determinant of the transpose equals the determinant itself.
- adj(A) × A = A × adj(A) = |A| × I, where I is the identity matrix.
- A⁻¹ = (1/|A|) × adj(A), provided |A| ≠ 0.
- Cramer's rule for two variables: if ax + by = e and cx + dy = f, then x = (ed − bf)/(ad − bc), y = (af − ec)/(ad − bc), when ad − bc ≠ 0.
Worked examples
Example 1: Evaluate the determinant of a 2 × 2 matrix
Find the determinant of A = [[5, 3], [2, 4]].
Solution: Using the formula det(A) = ad − bc, we have a = 5, b = 3, c = 2, d = 4. det(A) = 5 × 4 − 3 × 2 = 20 − 6 = 14.
Example 2: Evaluate a 3 × 3 determinant using cofactor expansion
Find |B| where B = [[2, 0, 1], [3, 1, 2], [1, 4, 3]].
Solution: Expand along the first row. Cofactor of 2: C₁₁ = (+1) × det([[1, 2], [4, 3]]) = 1 × 3 − 2 × 4 = 3 − 8 = −5. Cofactor of 0: C₁₂ = (−1) × det([[3, 2], [1, 3]]) = (−1)(9 − 2) = −7. But the element is 0, so contribution is 0. Cofactor of 1: C₁₃ = (+1) × det([[3, 1], [1, 4]]) = 12 − 1 = 11.
B| = 2 × (−5) + 0 × (−7) + 1 × 11 = −10 + 0 + 11 = 1.
Example 3: Find the area of a triangle using determinants
A triangle has vertices P(1, 2), Q(4, 6), R(3, 1). Find its area.
Solution: Area = (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| Substituting: = (1/2)|1(6 − 1) + 4(1 − 2) + 3(2 − 6)| = (1/2)|1 × 5 + 4 × (−1) + 3 × (−4)| = (1/2)|5 − 4 − 12| = (1/2)|−11| = 11/2 = 5.5 square units.
Common mistakes
- Forgetting the sign pattern in cofactors → remember C_ij = (−1)^(i+j) × M_ij; the checkerboard starts with + at position (1,1).
- Multiplying all elements by k and thinking |kA| = k × |A| → for an n × n matrix, |kA| = k^n × |A|.
- Adding a multiple of one row to another and also multiplying the determinant by that factor → this operation leaves the determinant unchanged; only multiplying a single row by k changes the value.
- Confusing the adjoint with the inverse → adj(A) is the transpose of the cofactor matrix; the inverse additionally divides by |A|.
- Using Cramer's rule when the coefficient determinant is zero → the rule fails; the system has either no solution or infinitely many.
Quick revision
- Determinant of a 2 × 2: ad − bc; of a 3 × 3: expand using cofactors along any row or column.
- Swapping two rows changes the sign; identical rows give zero determinant.
- A matrix is invertible exactly when its determinant is non-zero.
- Inverse formula: A⁻¹ = (1/|A|) × adj(A).
- Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) uses a determinant formula and equals half the absolute value.