Inverse matrices

Definition

For a square matrix A, the inverse matrix A⁻¹ is a matrix such that:

  • A⁻¹A = AA⁻¹ = I (where I is the identity matrix)

A matrix has an inverse if and only if it is non-singular (i.e., det(A) ≠ 0).

Properties of Inverse Matrices

  1. Uniqueness: If A is invertible, its inverse A⁻¹ is unique.

  2. Inverse of product: (AB)⁻¹ = B⁻¹A⁻¹ (Note the order reversal)

  3. Inverse of transpose: (A^T)⁻¹ = (A⁻¹)^T

  4. Inverse of inverse: (A⁻¹)⁻¹ = A

  5. Determinant relationship: det(A⁻¹) = 1/det(A)

  6. Scalar multiplication: (kA)⁻¹ = (1/k)A⁻¹ for k ≠ 0

  7. Invertibility criteria:

    • A is invertible iff det(A) ≠ 0
    • A is invertible iff rank(A) = n (for an n×n matrix)
    • A is invertible iff it has n linearly independent rows/columns
    • A is invertible iff the homogeneous system Ax = 0 has only the trivial solution

Methods for Finding the Inverse

1. Adjoint Method (for smaller matrices)

For an n×n matrix A:

  • A⁻¹ = (1/det(A)) × adj(A)

Where adj(A) is the adjoint (or adjugate) of A, a matrix whose (i,j) entry is:

  • adj(A)ᵢⱼ = (-1)^(i+j) × Mⱼᵢ

Where Mⱼᵢ is the determinant of the submatrix formed by removing row j and column i from A.

Note that for the adjoint, the indices are swapped (this is equivalent to finding the cofactor matrix and then transposing it).

2. Gauss-Jordan Elimination (preferred for larger matrices)

  1. Form the augmented matrix [A I]
  2. Apply row operations to transform A into the identity matrix I
  3. The result will be [I A⁻¹]

3. Formula for 2×2 Matrices

For A = $\begin{bmatrix} a & b \ c & d \end{bmatrix}$ with det(A) = ad - bc ≠ 0:

A⁻¹ = $\frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$

Examples

Example 1: 2×2 Matrix

For A = $\begin{bmatrix} 4 & -3 \ 2 & -1 \end{bmatrix}$:

  1. Calculate det(A) = 4×(-1) - (-3)×2 = -4 - (-6) = 2 ≠ 0, so A is invertible.

  2. Using the formula for 2×2 matrices: A⁻¹ = $\frac{1}{2} \begin{bmatrix} -1 & 3 \ -2 & 4 \end{bmatrix} = \begin{bmatrix} -1/2 & 3/2 \ -1 & 2 \end{bmatrix}$

  3. Verify: A×A⁻¹ = $\begin{bmatrix} 4 & -3 \ 2 & -1 \end{bmatrix} \times \begin{bmatrix} -1/2 & 3/2 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ = I

Example 2: 3×3 Matrix Using Gauss-Jordan

For A = $\begin{bmatrix} 1 & 2 & 0 \ 3 & 5 & 1 \ 5 & -8 & -3/4 \end{bmatrix}$:

  1. Form the augmented matrix [A I]:    
    $\begin{bmatrix} 1 & 2 & 0 & & 1 & 0 & 0 \ 3 & 5 & 1 & & 0 & 1 & 0 \ 5 & -8 & -3/4 & & 0 & 0 & 1 \end{bmatrix}$
  2. Apply row operations to get I on the left side:
    • R₂ = R₂ - 3R₁
    • R₃ = R₃ - 5R₁
    • …continue with other row operations…

  3. After completing all row operations, the result will be: $\begin{bmatrix} 1 & 0 & 0 & | & a & b & c \ 0 & 1 & 0 & | & d & e & f \ 0 & 0 & 1 & | & g & h & i \end{bmatrix}$

  4. The inverse is: A⁻¹ = $\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$

Example 3: Non-invertible Matrix

For A = $\begin{bmatrix} 1 & -3 \ 2 & -6 \end{bmatrix}$:

  1. Calculate det(A) = 1×(-6) - (-3)×2 = -6 + 6 = 0

  2. Since det(A) = 0, A is singular (non-invertible) and has no inverse.

Special Cases

1. Diagonal Matrices

For a diagonal matrix D = diag(d₁, d₂, …, dₙ) with all dᵢ ≠ 0:

  • D⁻¹ = diag(1/d₁, 1/d₂, …, 1/dₙ)

2. Triangular Matrices

For a triangular matrix (upper or lower) with all diagonal elements non-zero:

  • The inverse is also triangular (of the same type)
  • Diagonal elements of the inverse are reciprocals of the original diagonal elements

3. Elementary Matrices

Elementary matrices (representing a single row operation) are always invertible:

  • Row swap: its own inverse
  • Row multiplication by k ≠ 0: inverse multiplies the same row by 1/k
  • Row addition of multiple of another row: inverse subtracts the same multiple

Computational Techniques

Method Selection

  1. For 2×2 matrices, use the direct formula
  2. For 3×3 matrices, either:
    • Use the adjoint method if exact fractions are needed
    • Use Gauss-Jordan elimination for numerical efficiency
  3. For larger matrices, always use Gauss-Jordan elimination

Verification

Always verify your answer by checking:

  • A×A⁻¹ = I
  • A⁻¹×A = I

Common Mistakes to Avoid

  1. Forgetting to check if det(A) = 0
  2. Arithmetic errors during Gauss-Jordan elimination
  3. Not completing the full row reduction (stopping at row echelon form instead of reduced row echelon form)
  4. Confusing the order of operations in (AB)⁻¹

Exercise Strategies

For matrices with specific patterns:

  1. Upper triangular matrix:
    • Check if diagonal elements are non-zero
    • Inverse is also upper triangular
    • Use back-substitution with [A I]
  2. Matrices with many zeros:
    • Choose the method that takes advantage of the sparsity pattern
  3. Symbolic matrices (with parameters):
    • Find conditions for invertibility by calculating determinant
    • Solve for the parameter values where det(A) = 0
    • For valid parameter values, find the inverse symbolically

      Exercises

  • Find, if there exists, the inverse of the following matrices:
    • $A = \begin{pmatrix} 4 & -3 \ 2 & -1 \end{pmatrix}$
    • $A = \begin{pmatrix} 1 & -3 \ 2 & -6 \end{pmatrix}$
    • $A = \begin{pmatrix} -1 & 2 & -4 \ 3 & -5 & 6 \ -4 & 3 & -7 \end{pmatrix}$
    • $A = \begin{pmatrix} 11 & 2 & -3 \ 3 & 5 & 1 \ 5 & -8 & -5 \end{pmatrix}$
    • $A = \begin{pmatrix} 1 & 2 & 0 \ 3 & 5 & 1 \ 5 & -8 & -\frac{3}{4} \end{pmatrix}$
    • $A = \begin{pmatrix} -2 & 0 & 3 & 1 \ -3 & -1 & 2 & 4 \ 0 & 2 & 1 & -2 \ 3 & 1 & 2 & -5 \end{pmatrix}$
    • $A = \begin{pmatrix} 0 & 0 & -1 \ 1 & 2 & 3 \ 1 & 0 & 1 \end{pmatrix}$