Determinants and Inverse Matrices
The determinant
The determinant of a square matrix is a single number with important geometric and algebraic meaning. For a 2×2 matrix:
A = [a b; c d] det A = |A| = ad − bcWhat the determinant tells you
- The determinant is the area scale factor of the transformation the matrix represents (in 3D, the volume scale factor).
- A negative determinant means the transformation includes a reflection (orientation reversed).
- If det A = 0, the matrix is singular — it has no inverse, and it maps the plane onto a line (area scale factor 0).
The inverse matrix
The inverse A⁻¹ "undoes" A: A A⁻¹ = A⁻¹A = I. For a 2×2 matrix (provided det A ≠ 0):
A⁻¹ = (1/det A) [d −b; −c −a... ]More precisely: swap the leading-diagonal elements, change the sign of the other two, then divide by the determinant:
A⁻¹ = (1/(ad−bc)) [d −b; −c a]Solving simultaneous equations with matrices
A system like ax + by = e, cx + dy = f can be written as AX = B, where A is the coefficient matrix. The solution is:
X = A⁻¹BThis works only if det A ≠ 0 (a unique solution). If det A = 0, the equations have no unique solution (no solution, or infinitely many).
Order of an inverse product
(AB)⁻¹ = B⁻¹A⁻¹ (note the reversed order)Worked example
Find the inverse of A = [2 1; 3 2].
- det A = (2×2) − (1×3) = 4 − 3 = 1.
- A⁻¹ = (1/1)[2 −1; −3 2] = [2 −1; −3 2]. (Swap 2 and 2, negate 1 and 3.) ✓
Common mistakes
- Computing det A as ad + bc instead of ad − bc.
- Forgetting to divide by the determinant when finding the inverse.
- Trying to invert a singular matrix (det = 0 has no inverse).
Exam tips
- Always find the determinant first; if it's 0, state the matrix is singular (no inverse).
- For the 2×2 inverse: swap the diagonal, negate the off-diagonal, divide by det.
- Solve linear systems with X = A⁻¹B.
Key facts to remember
- det [a b; c d] = ad − bc = area scale factor; det = 0 ⇒ singular (no inverse).
- 2×2 inverse: A⁻¹ = (1/det A)[d −b; −c a] (swap diagonal, negate off-diagonal).
- Solve AX = B with X = A⁻¹B; (AB)⁻¹ = B⁻¹A⁻¹.