Inverse of a Matrix [ English ]

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1. Introduction

In matrix theory, the inverse of a matrix plays a role similar to the reciprocal of a number. It is used to:

• Solve systems of linear equations
• Reverse matrix transformations
• Perform advanced computations in mathematics and computer science

Only certain matrices have an inverse—specifically, non-singular matrices.

2. Definition of Inverse Matrix

The inverse of a matrix A is another matrix A⁻¹ such that:

A × A⁻¹ = A⁻¹ × A = I

Where:

• I is the identity matrix

3. Condition for Existence

A matrix has an inverse if and only if:

det(A) ≠ 0

✔ Such a matrix is called non-singular ❌ If det(A) = 0, the matrix is singular and has no inverse

4. Inverse of a 2×2 Matrix

Let:

A = | a  b |
    | c  d |

Step 1: Compute Determinant

det(A) = ad − bc

Step 2: Apply Formula

A⁻¹ = (1 / (ad − bc)) × |  d  -b |
                         | -c   a |

Example

Let:

A = | 2  3 |
    | 1  4 |

Step 1:

det(A) = (2×4 − 3×1) = 8 − 3 = 5

Step 2:

A⁻¹ = (1/5) × | 4  -3 |
               | -1  2 |

5. Inverse of a 3×3 Matrix

For a 3×3 matrix, the inverse is found using:

A⁻¹ = (1 / det(A)) × adj(A)

Steps:

1. Find determinant of A
2. Find cofactor matrix
3. Take transpose → adjoint
4. Multiply by 1/det(A)

Example

Let:

A = | 1  2  3 |
    | 0  4  5 |
    | 1  0  6 |

Step 1: Determinant

det(A) = 1(4×6 − 5×0) − 2(0×6 − 5×1) + 3(0×0 − 4×1)
       = 24 + 10 − 12
       = 22

Step 2: Adjoint (from previous result)

adj(A) = | 24  -12  -2 |
         |  5    3  -5 |
         | -4    2   4 |

Step 3: Inverse

A⁻¹ = (1/22) × adj(A)

6. Properties of Inverse

(a) Inverse of Identity

I⁻¹ = I

(b) Inverse of Product

(AB)⁻¹ = B⁻¹ A⁻¹

(c) Inverse of Transpose

(Aᵀ)⁻¹ = (A⁻¹)ᵀ

(d) Inverse of Inverse

(A⁻¹)⁻¹ = A

7. Applications of Inverse

(a) Solving Linear Equations

AX = B  ⇒  X = A⁻¹B

(b) Computer Science

• Graphics transformations
• Cryptography

(c) Mathematics

• Algebraic computations
• Matrix theory

8. Key Observations

• Only square matrices can have inverses
• Determinant must be non-zero
• Inverse is unique

9. Summary

• The inverse of A is denoted by A⁻¹
• Defined by: A × A⁻¹ = I
• Exists only if det(A) ≠ 0
• For 2×2: direct formula
• For 3×3: use A⁻¹ = (1/det(A)) × adj(A)