Matrix Inverse

Subtopic: Matricies

The inverse of a matrix A is the matrix (A^-1) that "undoes" A: multiplying gives the identity A*(A^-1)=(A^-1)*A=I. Not every matrix has an inverse—only square matrices with nonzero determinant. Computing inverses is essential for solving linear systems and understanding linear transformations.

Introduction

For numbers, the inverse of 3 is 1/3: multiply them and you get 1. For matrices, we seek (A^-1) such that multiplying gives the identity matrix (the matrix equivalent of 1).

But unlike numbers, not every nonzero matrix has an inverse. A matrix that collapses space (determinant zero) can't be undone—information is lost.

Definition

A square matrix A is invertible (nonsingular) if there exists a matrix (A^-1) such that:

A*A^(−1)=A^(−1)*A=I

The inverse is unique when it exists. A matrix without an inverse is called singular.

2×2 Inverse Formula

For a 2⨉2 matrix:

([a,b],[c,d])^(−1)=1/(a*d−b*c)*([d,−b],[−c,a])

Swap the diagonal entries, negate the off-diagonal entries, divide by the determinant.

This only works when det()=a*d-b*c≠0.

Worked Example

Find the inverse of:

A=([4,7],[2,6])

Step 1: Compute determinant = 4*(6) -7*(2)=24-14=10≠0

Step 2: Apply formula:

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

Step 3: Verify:

A*A^(−1)=([4,7],[2,6])*([0.6,−0.7],[−0.2,0.4])=([1,0],[0,1])

General Method: Gauss-Jordan

For larger matrices, augment [[A,I]] and row reduce to [[I,A^(-1)]]:

1. Write the augmented matrix [[A,I]]

2. Apply row operations to transform the left side to I

3. The right side becomes A^(-1)

If at any point a row of the left side becomes all zeros, A is singular (no inverse).

When Does the Inverse Exist?

A square matrix A is invertible if and only if any of these equivalent conditions holds:

• det(A)≠0

• A has full rank (rank =n for an n×n matrix)

• The columns of A are linearly independent

• The rows of A are linearly independent

• The null space of A is {0}

• 0 is not an eigenvalue of A

• A*x=b has a unique solution for every b

Key Properties

• (A^(-1))^(−1)=A

• (A*B)^(−1)=B^(-1)*A^(-1) (reverse order!)

• (A^T)^(−1)=(A^(−1))^T

• (c*A)^(−1)=1/c*A^(−1)

• det(A^(−1))=1/det(A)

Solving Linear Systems

If A is invertible, the system A*x=b has unique solution:

x=A^(−1)*b

However, computing A^(-1) explicitly is usually inefficient. For solving systems, LU decomposition or direct methods are preferred.

Geometric Interpretation

If A represents a transformation (rotation, scaling, shear), then A^(−1) represents the reverse transformation.

• Rotation by θ has inverse rotation by −θ

• Scaling by 2 has inverse scaling by 1/2

• A projection (collapsing a dimension) has no inverse — you cannot recover the lost information

Applications

In cryptography, encryption matrices must be invertible to allow decryption.

In computer graphics, applying a transformation then its inverse returns to the original state.

In control theory, system invertibility determines whether inputs can achieve arbitrary outputs.

Summary

The inverse A^(−1) undoes A: A*A^(−1)=I. It exists if and only if det(A)≠0 (equivalently: full rank, independent columns, no zero eigenvalue). For 2×2 matrices: swap the diagonal entries, negate the off-diagonal entries, and divide by det(A). For larger matrices: use Gauss–Jordan elimination. Key identity: (A*B)^(−1)=B^(−1)*A^(−1) (reverse order). The inverse reverses the geometric transformation.