Bonggot- The Inverse of a Matrix - Final ( LINEAR ALGEBRA )

The Inverse of a Matrix Linear Algebra

Definition of the Inverse of a Matrix

  • A matrix A is invertible (or nonsingular) if there exists a matrix B such that:

    • AB = I

    • BA = I

  • I is the identity matrix of order n.

  • The matrix B is called the multiplicative inverse of A.

  • A matrix that does not have an inverse is referred to as noninvertible (or singular).

Uniqueness of an Inverse Matrix

Theorem 2.7

  • If A is an invertible matrix, then its inverse is unique: A⁻¹.

  • Non-square matrices do not have inverses due to dimensionality issues.

  • If A is m × n and B is n × m (where m ≠ n), then AB and BA will have different sizes, hence cannot equal each other.

  • Not all square matrices have inverses, but if a square matrix does have an inverse, then this inverse is unique.

Proof of Uniqueness

  • Assume A is invertible with inverse B.

  • If there exists another inverse C such that AC = I and CA = I, then prove B = C as follows:

    • Start with AB = I

    • Multiply by C: C(AB) = CI

    • Simplifies to (CA)B = C

    • Therefore, B = C implies uniqueness of the inverse.

  • Thus, the inverse of a matrix A, denoted A⁻¹, is unique.

Finding the Inverse of a Matrix

Example 1: Verification of Inverse

  • To show that matrix B is the inverse of A, verify that AB = I and BA = I.

Example 2: Compute inverse by Solution

  • To find A⁻¹, solve the equation AX = I for X.

  • Transform the equation into a system of linear equations by equating corresponding entries.

  • Example matrices provided and corresponding equations listed.

Example 2: Solving the Systems

  • After breaking down the equations, results in values for a₁, b₁, a₂, and b₂.

  • Confirm the inverse by checking with matrix multiplication.

  • Introduces Gauss-Jordan elimination method as a systematic approach to find the inverse.

Gauss-Jordan Elimination Steps

  1. Form matrix [A | I] and apply row operations.

  2. If successful, the resulting matrix will be [I | A⁻¹].

  3. If the row reduction cannot achieve IV, matrix A is noninvertible.

Example 3: Apply Methods

  • Use row operations on the matrix with identity adjunction to find A⁻¹.

  • Detailed steps of operations listed leading to the inverse.

Properties of Inverses

  • If A is an invertible matrix, and k is a nonzero scalar:

  1. (A⁻¹)⁻¹ = A

  2. (kA)⁻¹ = (1/k)A⁻¹

  3. The inverse of a product of two matrices:

    • (AB)⁻¹ = B⁻¹A⁻¹

  • Generalized for multiple matrices:

    • (A₁A₂...Aₖ)⁻¹ = Aₖ⁻¹Aₖ⁻²...A₁⁻¹

Cancellation Properties

Theorem 2.10

  • If C is invertible, then:

  1. Right cancellation: If CA = CB, then A = B.

  2. Left cancellation: If AC = BC, then A = B.

System of Equations with Unique Solutions

Theorem 2.11

  • For an invertible matrix A, the system of linear equations Ax = b has a unique solution represented by x = A⁻¹b.

Example 8: Using Inverses to Solve Systems

  • Systems of equations presented with coefficient matrix A.

  • Solve using the inverse matrix A⁻¹.

  • Solutions for each example outlined with details on computation achieved via A⁻¹.

Conclusion

  • Summarizes important aspects around matrix inverses and their properties, significance, and utility in solving equations.