Introduction
In linear algebra, the inverse of a matrix plays a role similar to the reciprocal of a number. For a non-zero number a, the reciprocal 1/a satisfies a × (1/a) = 1. Likewise, for a square matrix A, its inverse matrix A⁻¹ satisfies: A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix of the same order as A.
What is a Non-Singular Matrix?
A square matrix is called non-singular if its determinant is not zero.
If det(A) ≠ 0, then A is non-singular and has an inverse.
If det(A) = 0, then A is singular and has no inverse.
Condition for Existence of Inverse
A square matrix A has an inverse if and only if det(A) ≠ 0. This means only non-singular matrices have inverses.
Formula for the Inverse of a Matrix
For a 2 × 2 matrix A = [[a, b], [c, d]], the inverse is given by:
A⁻¹ = (1 / (ad – bc)) * [[d, -b], [-c, a]]
Note: The determinant ad – bc must not be zero.
General Formula (Using Adjoint)
For a square matrix A: A⁻¹ = (1 / |A|) Adj(A)
where |A| is the determinant of A and Adj(A) is the adjugate (adjoint) of A.
Steps to Find the Inverse of a Matrix
- Find the determinant |A|.
- Find the cofactor matrix of A.
- Transpose the cofactor matrix to get the adjoint matrix.
- Divide the adjoint by the determinant to get A⁻¹ = (1 / |A|) Adj(A).
Example
Find the inverse of A = [[2, 1], [5, 3]]
Step 1: |A| = (2×3) – (1×5) = 1
Step 2: Adj(A) = [[3, -1], [-5, 2]]
Step 3: A⁻¹ = 1/1 × [[3, -1], [-5, 2]] = [[3, -1], [-5, 2]]
Verification
A × A⁻¹ = [[2, 1], [5, 3]] × [[3, -1], [-5, 2]] = [[1, 0], [0, 1]]
Hence, A⁻¹ is correct.
Applications of Matrix Inverse
- Solving systems of linear equations (AX = B ⇒ X = A⁻¹B)
- Computer graphics transformations
- Cryptography and encoding
- Engineering simulations and data modeling
Summary
Concept Key Idea Non-Singular Matrix Determinant ≠ 0 Singular Matrix Determinant = 0 Inverse Formula A⁻¹ = (1 / |A|) Adj(A) Condition Only non-singular matrices have inverses Conclusion
The inverse of a non-singular square matrix is a powerful concept that simplifies solving equations and performing matrix operations. Remember — only matrices with a non-zero determinant can be inverted