Concept of Inverse
For a square matrix A of order n, another matrix B is called the inverse of A if:
A · B = B · A = I
where I is the identity matrix of order n. We write B = A⁻¹.
⚠️ Condition: The inverse of A exists if and only if |A| ≠ 0. A matrix with |A| = 0 is called singular and has no inverse.
Always check |A| ≠ 0 before attempting to find the inverse. This is Step 0 in every inverse problem.
The Inverse Formula
This formula is derived from the fundamental result: A · adj(A) = |A| · I
Dividing both sides by |A|: A · (adj(A)/|A|) = I, which means A⁻¹ = adj(A)/|A|
For a 2×2 matrix A = [[a,b],[c,d]], the inverse has a shortcut:
A⁻¹ = (1/|A|) × [[d, −b],[−c, a]]
Just swap the diagonal elements and negate the off-diagonal elements, then divide by |A|.
Step-by-Step Method
Check |A| ≠ 0
Compute the determinant first. If |A| = 0, inverse does not exist — stop.
Find all Cofactors Aᵢⱼ
Compute the cofactor for every element using (−1)^(i+j) × Mᵢⱼ.
Form the Cofactor Matrix
Arrange all cofactors into a matrix at their corresponding positions.
Transpose → get adj(A)
Take the transpose of the cofactor matrix to get adj(A).
Divide by |A|
A⁻¹ = (1/|A|) × adj(A). Multiply each element of adj(A) by 1/|A|.
Verify: A · A⁻¹ = I
Multiply A by A⁻¹ and confirm you get the identity matrix.
Inverse of a 2×2 Matrix
|A| = (3)(2) − (1)(5) = 6 − 5 = 1 ≠ 0 ✓ Inverse exists
adj(A) = [[2, −1],[−5, 3]] (swap diagonal, negate off-diagonal)
A⁻¹ = (1/1) × [[2,−1],[−5,3]] = [[2,−1],[−5,3]]
✅ Verify: A × A⁻¹ = [[3,1],[5,2]] × [[2,−1],[−5,3]] = [[6−5, −3+3],[10−10, −5+6]] = [[1,0],[0,1]] = I ✓
Inverse of a 3×3 Matrix
Expand along R1: 1·|3,0;2,−1| − 0 + 0 = 1·(−3−0) = −3 ≠ 0 ✓
(+1)|3,0;2,−1| = −3
(−1)|3,0;5,−1| = −(−3−0) = 3
(+1)|3,3;5,2| = 6−15 = −9
(−1)|0,0;2,−1| = −(0−0) = 0
(+1)|1,0;5,−1| = −1−0 = −1
(−1)|1,0;5,2| = −(2−0) = −2
(+1)|0,0;3,0| = 0−0 = 0
(−1)|1,0;3,0| = −(0−0) = 0
(+1)|1,0;3,3| = 3−0 = 3
Cofactor matrix transposed:
adj(A) = [[−3, 0, 0],[3, −1, 0],[−9, −2, 3]]
A⁻¹ = (1/−3) × adj(A) = [[1, 0, 0],[−1, 1/3, 0],[3, 2/3, −1]]
Important Results & Properties
• (A⁻¹)⁻¹ = A — Inverse of inverse is the original matrix
• (AB)⁻¹ = B⁻¹A⁻¹ — Reversal rule for products
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ — Inverse of transpose
• |A⁻¹| = 1/|A| — Determinant of inverse
• A·A⁻¹ = A⁻¹·A = I — Definition check
| Matrix Type | Condition | Inverse Exists? |
|---|---|---|
| Non-Singular | |A| ≠ 0 | Yes ✓ |
| Singular | |A| = 0 | No ✗ |
| Identity I | |I| = 1 | Yes (I⁻¹ = I) |
| Scalar Matrix kI | k ≠ 0 | Yes ((kI)⁻¹ = (1/k)I) |