Determinants and inverses
Objective. Test invertibility and calculate inverses.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The determinant of a 2×2 matrix A = [[a,b],[c,d]] is det(A) = ad − bc, and A is invertible precisely when det(A) ≠ 0. When invertible, the inverse is A⁻¹ = (1/det(A))[[d,−b],[−c,a]], swapping the leading diagonal and negating the other. A zero determinant means the matrix maps the plane onto a line or point, collapsing area, so no inverse can recover the original points. For example, A = [[2,1],[1,1]] has det(A) = 2−1 = 1, so A⁻¹ = [[1,−1],[−1,2]].
Worked example
Find the determinant and, if it exists, the inverse of the matrix M = [[3,2],[4,3]].
- 1det(M) = (3×3) − (2×4) = 9 − 8 = 1
- 2Since det(M) = 1 ≠ 0, M is invertible
- 3Swap diagonal entries and negate off-diagonal: [[3,−2],[−4,3]]
- 4Divide by det(M) = 1, so M⁻¹ = [[3,−2],[−4,3]]
Answer: det(M) = 1; M⁻¹ = [[3,−2],[−4,3]]
Concept mastery
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Practice
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Start practisingCommon mistake. Students forget to divide by the determinant, writing the inverse as just [[d,−b],[−c,a]]. Always check that A × A⁻¹ gives the identity matrix as a quick verification.
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