How do I find the inverse of a 2×2 matrix?

Compute the determinant ad − bc. If it is not zero, swap a and d, change the signs of b and c, and divide every entry by the determinant.

Why is AB not equal to BA?

Matrix multiplication combines rows of the first matrix with columns of the second, so swapping the order changes which rows meet which columns. Only special pairs, such as a matrix and its inverse, commute.

Are eigenvalues in the DSE M2 syllabus?

No. M2 matrices cover operations, the transpose, and inverses of square matrices of order 2 and 3. Eigenvalues and diagonalisation are not listed in the curriculum.

Matrices and Inverse Matrices (DSE M2)

Matrices are multiplied row by column, and in general AB ≠ BA. A square matrix A has an inverse exactly when |A| ≠ 0; then A⁻¹ = (1/|A|) adj A, which for a 2 × 2 matrix [a b; c d] is (1/(ad − bc)) [d −b; −c a].

Solve Matrices questions with AI

Enrichment (not examined): the matrix [2 1; 1 2] sends the unit square to a parallelogram of area 3, which is its determinant. A determinant of 0 would flatten the square to a line, which is why such a matrix has no inverse. Try it yourself →

Matrix Formulas (DSE M2)

Inverse of a 2 × 2 matrix[a b; c d]⁻¹ = (1/(ad − bc)) [d −b; −c a]

Inverse in generalA⁻¹ = (1/|A|) adj A, |A| ≠ 0

Not commutativeAB ≠ BA in general

Inverse of a product(AB)⁻¹ = B⁻¹A⁻¹

Determinant of a product|AB| = |A||B|, |A⁻¹| = 1/|A|

Transpose and powers(Aᵀ)⁻¹ = (A⁻¹)ᵀ, (Aⁿ)⁻¹ = (A⁻¹)ⁿ

Scalar multiple(λA)⁻¹ = λ⁻¹A⁻¹

How to Find the Inverse of a 3 × 3 Matrix

Find |A|. If it is 0, stop: there is no inverse.
For each entry, find its cofactor: delete the entry’s row and column, take the 2 × 2 determinant, and apply the sign pattern + − + / − + − / + − +.
Transpose the matrix of cofactors to get adj A.
A⁻¹ = (1/|A|) adj A.
Check by multiplying: AA⁻¹ must be the identity matrix.

Elementary row operations on [A | I] are an equally acceptable method.

Worked Examples — Matrices

Example 1: Inverse of a 2 × 2 matrix

Find the inverse of A = [2 1; 5 3].

|A| = (2)(3) − (1)(5) = 1 ≠ 0
Swap the diagonal entries and change the sign of the other two: [3 −1; −5 2]
A⁻¹ = (1/1) [3 −1; −5 2]
Check: [2 1; 5 3][3 −1; −5 2] = [1 0; 0 1] ✓

Answer: A⁻¹ = [3 −1; −5 2]

Example 2: "Hence find the inverse" from a matrix equation

Let A = [2 1; 5 3]. Show that A² − 5A + I = 0. Hence find A⁻¹ in terms of A and I.

A² = [2 1; 5 3][2 1; 5 3] = [9 5; 25 14]
5A − I = [10 5; 25 15] − [1 0; 0 1] = [9 5; 25 14] = A²
So A² − 5A + I = 0, i.e. A(5I − A) = I
A⁻¹ = 5I − A = [3 −1; −5 2]

Answer: A⁻¹ = 5I − A

💡 Exam Traps & Secrets

⚠️ Order of multiplicationAB and BA are usually different. When you multiply both sides of an equation by A⁻¹, multiply on the same side of both: AX = B gives X = A⁻¹B, not BA⁻¹.

⚠️ "Hence" means use the equationIf the question says "hence find A⁻¹", factorise the matrix equation into A(…) = I. Using the 2 × 2 formula instead ignores the instruction and may score nothing.

⚠️ (AB)⁻¹ reverses the order(AB)⁻¹ = B⁻¹A⁻¹, not A⁻¹B⁻¹. Likewise (A + B)² ≠ A² + 2AB + B² unless AB = BA.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(2 × 2 inverse)Find the inverse of the matrix [3 1; 4 2].

(3 × 3 inverse)Find the inverse of the 3×3 matrix with rows (1, 0, 2), (0, 1, 1), (1, 1, 0).

(Induction with matrices)Let A = [1 1; 0 1]. Prove by mathematical induction that Aⁿ = [1 n; 0 1] for all positive integers n.