Inverse Matrix

1. What Is an Inverse Matrix?

The inverse of a square matrix A, written A⁻¹, is the matrix that “undoes” A. Multiplying a matrix by its inverse — in either order — gives the identity matrix I:

A A⁻¹ = A⁻¹ A = I

The inverse plays the role that division plays for ordinary numbers: it is how you isolate x in the matrix equation Ax = b. Crucially, not every matrix has one. An inverse exists only when the matrix is square and its determinant is non-zero. A matrix with no inverse is called singular.

역행렬 A⁻¹은 A를 "되돌리는" 행렬로, 어느 순서로 곱해도 항등행렬 I가 됩니다(A A⁻¹ = A⁻¹ A = I). 일반 수의 나눗셈에 해당하며, Ax = b에서 x를 구할 때 씁니다. 정사각이고 행렬식이 0이 아닐 때만 존재하며, 없는 경우 특이(singular) 행렬이라 합니다.

2. The 2×2 Inverse Formula

For A = [[a, b], [c, d]], swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant:

A⁻¹ = (1 / (ad − bc)) [[d, −b], [−c, a]]

The whole formula collapses the instant the determinant is 0 — division by zero — which is exactly why a singular matrix has no inverse.

A = [[a, b], [c, d]]의 역행렬은 대각 원소 a, d를 맞바꾸고, 비대각 원소 b, c의 부호를 바꾼 뒤 행렬식으로 나눕니다: A⁻¹ = (1/(ad − bc)) [[d, −b], [−c, a]]. 행렬식이 0이면 0으로 나누게 되어 역행렬이 없습니다.

3. Worked Example with Verification

For A = [[4, 7], [2, 6]], the determinant is 4·6 − 7·2 = 24 − 14 = 10 (non-zero, so the inverse exists). Apply the formula:

A⁻¹ = (1/10) [[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]

Verify by multiplying A A⁻¹:

• (1,1): 4(0.6) + 7(−0.2) = 2.4 − 1.4 = 1
• (1,2): 4(−0.7) + 7(0.4) = −2.8 + 2.8 = 0
• (2,1): 2(0.6) + 6(−0.2) = 1.2 − 1.2 = 0
• (2,2): 2(−0.7) + 6(0.4) = −1.4 + 2.4 = 1

The product is [[1, 0], [0, 1]] = I ✓.

A = [[4, 7], [2, 6]]의 행렬식은 10이므로 역행렬이 존재합니다. A⁻¹ = (1/10)[[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]. 검산하면 A A⁻¹ = [[1, 0], [0, 1]] = I 로 맞습니다.

4. Larger Matrices & Key Properties

For 3×3 and larger, the clean approach is Gauss-Jordan elimination: write the augmented matrix [A | I] and row-reduce until the left block becomes the identity; the right block is then A⁻¹. (The adjugate-over-determinant formula also works but is slow.)

• Existence. A⁻¹ exists ⇔ det(A) ≠ 0 ⇔ A has full rank.
• Product rule. (AB)⁻¹ = B⁻¹A⁻¹ — the order reverses, like the transpose.
• Double inverse. (A⁻¹)⁻¹ = A.
• Transpose. (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
• Determinant. det(A⁻¹) = 1 / det(A).

3×3 이상은 가우스–조던 소거법이 깔끔합니다: [A | I]를 행 줄여 왼쪽이 I가 되면 오른쪽이 A⁻¹입니다. 성질: A⁻¹ 존재 ⇔ det(A) ≠ 0 ⇔ 풀계수, (AB)⁻¹ = B⁻¹A⁻¹, (A⁻¹)⁻¹ = A, (Aᵀ)⁻¹ = (A⁻¹)ᵀ, det(A⁻¹) = 1/det(A).

5. Frequently Asked Questions

When does a matrix have an inverse? Only when it is square and its determinant is non-zero. Such a matrix is called invertible (or non-singular).

Why does a singular matrix have no inverse? Its determinant is 0, so the inverse formula divides by zero. Geometrically the matrix collapses space and cannot be undone.

How do you invert a 2×2 matrix? Swap the diagonal entries, negate the off-diagonal entries, and divide everything by the determinant ad − bc.

정사각이고 행렬식이 0이 아닐 때만 역행렬이 있습니다(가역·비특이 행렬). 특이 행렬은 행렬식이 0이라 0으로 나누게 되어 역행렬이 없습니다. 2×2는 대각 원소를 바꾸고 비대각 부호를 바꾼 뒤 ad − bc로 나눕니다.

Ready to practice? Drill inverse matrices and the rest of linear algebra on C:Matrix, or review the full matrix reference and the related determinant, matrix multiplication, and linear systems.

실전 연습은 C:Matrix에서, 전체 개념은 행렬 레퍼런스와 행렬식, 행렬 곱셈, 연립방정식 문서에서 이어집니다.