AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.11 — The Inverse and Determinant of a Matrix
Notes — Reversing a 2 × 2 Transformation
💡 Learning Objectives (4.11.A)
By the end of this lesson you will be able to:
- Compute the determinant of a 2 × 2 matrix
- Determine whether a matrix is invertible from its determinant
- Compute the inverse of an invertible 2 × 2 matrix
- Use matrix inverses to solve simple matrix equations
1. Determinant of a 2 × 2 Matrix
For A = [[a, b], [c, d]], the DETERMINANT is the single number:
det(A) = ad − bc
It's also written |A|. The determinant is positive, negative, or zero — and that sign carries geometric meaning (we'll see in Topic 4.12).
📘 Example — Compute determinants
- [[3, 5], [2, 4]]: det = 3·4 − 5·2 = 12 − 10 = 2
- [[6, 9], [2, 3]]: det = 6·3 − 9·2 = 18 − 18 = 0
- [[1, −2], [3, 4]]: det = 4 − (−6) = 10
2. The Inverse of a 2 × 2 Matrix
If A is a square matrix, its INVERSE A⁻¹ is the matrix such that:
A · A⁻¹ = A⁻¹ · A = I
For a 2 × 2 matrix A = [[a, b], [c, d]], the inverse exists when det(A) ≠ 0, and is given by:
A⁻¹ = (1 / det(A)) · [[d, −b], [−c, a]]
In words: SWAP the diagonal entries (a and d), NEGATE the off-diagonal entries (b and c), then divide everything by the determinant.
📘 Example — Compute the inverse
A = [[3, 5], [2, 4]]. Already computed det(A) = 2.
- Swap diagonal: 3 ↔ 4. Negate off-diagonal: 5 → −5, 2 → −2
- Result: (1/2) · [[4, −5], [−2, 3]] = [[2, −5/2], [−1, 3/2]]
- Verify A · A⁻¹: [[3, 5], [2, 4]] · [[2, −5/2], [−1, 3/2]] = [[6 − 5, −15/2 + 15/2], [4 − 4, −5 + 6]] = [[1, 0], [0, 1]] ✓
3. When the Inverse Doesn't Exist
If det(A) = 0, the formula divides by zero — the matrix has NO INVERSE. We say A is SINGULAR (or non-invertible). Geometrically, this happens when the rows or columns of A are scalar multiples of each other (linearly dependent).
📘 Example — A singular matrix
A = [[2, 4], [3, 6]]. Notice row 2 is 1.5 × row 1.
- det = 2·6 − 4·3 = 12 − 12 = 0
- A is singular — no inverse exists.
4. Properties of Inverses
💡 Inverse Properties
- (A⁻¹)⁻¹ = A — inverting twice returns the original
- (AB)⁻¹ = B⁻¹A⁻¹ — note the order REVERSES
- (kA)⁻¹ = (1/k) A⁻¹ for k ≠ 0
- det(A⁻¹) = 1/det(A)
- If both A and B are invertible, so is AB
5. Solving Matrix Equations
Inverses let you solve equations like AX = B:
AX = B ⇒ X = A⁻¹ B (provided A⁻¹ exists)
⚠️ Order matters
Multiply on the LEFT by A⁻¹: A⁻¹AX = A⁻¹B ⇒ X = A⁻¹B. If you instead need to solve XA = B, multiply on the RIGHT: X = BA⁻¹.
📘 Example — Solve AX = B
A = [[2, 1], [1, 1]], B = [[3], [2]].
- det(A) = 2 − 1 = 1
- A⁻¹ = (1/1) · [[1, −1], [−1, 2]] = [[1, −1], [−1, 2]]
- X = A⁻¹B = [[1, −1], [−1, 2]] · [[3], [2]] = [[3 − 2], [−3 + 4]] = [[1], [1]]
6. Solving Systems of Linear Equations
A system of equations like {2x + y = 3, x + y = 2} can be written as AX = B, where:
A = [[2, 1], [1, 1]], X = [[x], [y]], B = [[3], [2]]
If A is invertible, the system has a UNIQUE solution X = A⁻¹B. If det(A) = 0, the system has either no solution or infinitely many solutions (depending on B).
7. Geometric Meaning of the Determinant
The absolute value of det(A) for a 2 × 2 matrix A is the AREA of the parallelogram formed by treating the columns of A as vectors emanating from the origin. The sign of the determinant tells you whether the columns are oriented in the standard counter-clockwise direction (positive) or reversed (negative).
If det = 0, the parallelogram has zero area — the two column vectors are parallel.
8. Summary
- det of [[a, b], [c, d]] is ad − bc
- Inverse exists iff det ≠ 0
- A⁻¹ = (1/det) · [[d, −b], [−c, a]]
- Use inverses to solve AX = B as X = A⁻¹B (left-multiply)
- (AB)⁻¹ = B⁻¹A⁻¹ — order reverses
- |det(A)| equals the area of the parallelogram of columns