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 = $\left[\begin{array}{ccc}a& bc& d\end{array}\right]$ with det(A) = ad - bc ≠ 0:

A⁻¹ = $\frac{1}{ad-bc}\left[\begin{array}{ccc}d& -b-c& a\end{array}\right]$

Examples

Example 1: 2×2 Matrix

For A = $\left[\begin{array}{ccc}4& -32& -1\end{array}\right]$:

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 = $\frac{1}{2}\left[\begin{array}{ccc}-1& 3-2& 4\end{array}\right]=\left[\begin{array}{ccc}-1/2& 3/2-1& 2\end{array}\right]$

3. Verify: A×A⁻¹ = $\left[\begin{array}{ccc}4& -32& -1\end{array}\right]\times \left[\begin{array}{ccc}-1/2& 3/2-1& 2\end{array}\right]=\left[\begin{array}{ccc}1& 00& 1\end{array}\right]$ = I

Example 2: 3×3 Matrix Using Gauss-Jordan

For A = $\left[\begin{array}{ccccccc}1& 2& 03& 5& 15& -8& -3/4\end{array}\right]$:

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: $\left[\begin{array}{ccccccccccccccccccc}1& 0& 0& |& a& b& c0& 1& 0& |& d& e& f0& 0& 1& |& g& h& i\end{array}\right]$

4. The inverse is: A⁻¹ = $\left[\begin{array}{ccccccc}a& b& cd& e& fg& h& i\end{array}\right]$

Example 3: Non-invertible Matrix

For A = $\left[\begin{array}{ccc}1& -32& -6\end{array}\right]$:

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:
  • $A=\left(\begin{array}{ccc}4& -32& -1\end{array}\right)$
  • $A=\left(\begin{array}{ccc}1& -32& -6\end{array}\right)$
  • $A=\left(\begin{array}{ccccccc}-1& 2& -43& -5& 6-4& 3& -7\end{array}\right)$
  • $A=\left(\begin{array}{ccccccc}11& 2& -33& 5& 15& -8& -5\end{array}\right)$
  • $A=\left(\begin{array}{ccccccc}1& 2& 03& 5& 15& -8& -\frac{3}{4}\end{array}\right)$
  • $A=\left(\begin{array}{ccccccccccccc}-2& 0& 3& 1-3& -1& 2& 40& 2& 1& -23& 1& 2& -5\end{array}\right)$
  • $A=\left(\begin{array}{ccccccc}0& 0& -11& 2& 31& 0& 1\end{array}\right)$