Inverse of a Matrix

The inverse of a matrix is what results in the identity matrix when multiplied by the original. Note that a matrix can have an inverse only if it has a non-zero determinant, and such matrices are called invertible matrices.

For example:

The inverse of a square matrix A, denoted as A⁻¹, is a matrix that, when multiplied with A, yields the identity matrix. If A is a square matrix, then its inverse satisfies \(A × A⁻¹ = A⁻¹ × A = I\)

\(A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \)

So, the inverse is:

\(A^{-1} = \begin{pmatrix} 4 & 3 \\ -1 & 2 \end{pmatrix} \div (2 \times 4 - 3 \times 1) = \begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix} \div 5 = \begin{pmatrix} \frac{4}{5} & -\frac{3}{5} \\ -\frac{1}{5} & \frac{2}{5} \end{pmatrix} \)
 

so, 

\(A \cdot A^{-1} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \times \begin{pmatrix} 4/5 & -3/5 \\ -1/5 & 2/5 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \)
 

and, 

\(A^{-1} \cdot A = \begin{pmatrix} 4/5 & -3/5 \\ -1/5 & 2/5 \end{pmatrix} \times \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \)

To calculate the inverse of a matrix, we need to learn certain terms related to it:

• Minor: To find the minor of an element in a matrix, remove its row and column. Then, calculate the determinant of the remaining smaller matrix.
  
  For example, in the matrix
  
  \(A = \begin{pmatrix} a11 & a12 & a13 \\ a21 & a22 & a23 \\ a31 & a32 & a33 \end{pmatrix} \)
  
  So, the minor of element a₁₁ is:
  
  \(\text{Minor of } a_{11} = \begin{vmatrix} a22 & a22 \\ a23 & a23 \end{vmatrix} \)

• Cofactor: To find the cofactor of an element, we multiply its minor by (-1)i + j.  Here, i and j denote the element’s row and column positions.  

• Determinant: A single value that represents a square matrix is referred to as the determinant. It can be calculated by expanding along any column or row. The determinant is the sum of each element multiplied by its cofactor.
  
  Determinant of A = a11 ​× Cofactor of a11 +a12 × Cofactor of a12​ +a13 × Cofactor of a13​. So, det(A) = a₁₁ × C₁₁ + a₁₂ × C₁₂ + a₁₃ × C₁₃.

• Singular Matrix: If the determinant of a matrix is zero (i.e., ∣A∣ = 0), the matrix is singular.  An inverse does not exist for a singular matrix.

• Non-Singular Matrix: A non-singular matrix has a non-zero determinant. (i.e.,  ∣A∣ ≠ 0). 

• Adjoint of a Matrix: The adjoint of a matrix can be determined by transposing the matrix of its cofactors.
   

We represent the inverse of any square matrix A as A–1. Additionally, the product of a matrix and its inverse always results in the identity matrix I.

The inverse of a square matrix A is given by the formula:

\(A–1= 1A× adj (A)\)

Where:
|A| → determinant of matrix A
Adj (A) → adjoint or transpose of cofactors of A

What are the properties of Inverse of a Matrix?

The inverse of the matrix has certain key features, as discussed below:
 

• The inverse of an inverse matrix is the original matrix.

(A^–1)^-1= A
 

• The inverse of the product of two matrices, A and B, is represented as.

\((AB)-1= B–1 A–1\)

 

• A matrix only possesses an inverse if it is non-singular.
  
   
• The inverse of the transpose of a matrix is equal to the transpose of its inverse: 

^{\( (A^{T})^{-1} = (A^{-1})^{T} \)}

How to Find the Matrix Inverse?

The two common methods to determine the inverse of a matrix are:

 

Using Matrix Formula

The inverse of a matrix is the adjoint divided by its determinant.

\( A^{-1} = \frac{\text{Adj } A}{|A|} \)
 

To find the matrix inverse, we use two Inverse matrix methods, such as:

• Determinant and adjoint Method
• Elementary Transformation Method
   

Determinant Method: The inverse of a matrix is calculated using the determinant and adjoint. The formula for a square matrix A:

\( A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A) \)

Where:

A^–1 → inverse of matrix A

det (A)→ determinant of A

adj(A) → adjoint of A

If the determinant of A is zero, the matrix has no inverse.

 

Elementary Transformation Method: In this method, we find the inverse using a step-by-step process:

Step 1: Express the matrix equation as [A\I]. Here, I is the identity matrix, which is in the same order as the matrix A.

Step 2: To convert matrix A into the identity matrix on the left, use only row operations or only column operations. Apply the same step to the identity matrix on the right side.
The matrix B on the right is the inverse of A once you obtain: \(I = B × A\).

Step 3: Use either row operations or column operations exclusively, not both simultaneously.
Using this method, we can quickly determine the inverse of a 2 × 2 matrix.

 

2 × 2 matrix Inverse


The inverse of a 2 × 2 matrix can be found using a simple method:

\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)


Let’s say: Since \( |A| = (ad - bc)\)

\(\text{Adj } A = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)

 
Using the formula:

\( A^{-1} = \frac{1}{|A|} \times \text{adj } A \)
 

So, 

\(A^{-1} = \frac{1}{(ad - bc)} \times \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)
 

Hence, the inverse of the 2 × 2 matrix is determined.


 

Inverse of a 3 × 3 Matrix


In the case of any 3 × 3 matrix:

\(A = \begin{pmatrix} x & y & z \\ l & m & n \\ a & b & c \end{pmatrix} \)

Use the inverse matrix formula:

\( A^{-1} = \frac{1}{|A|} \times \text{adj}(A) \)

Determinant of Inverse Matrix

The determinant of an inverse matrix is derived by taking the reciprocal of the original matrix’s determinant:

\( \det(A^{-1}) = \frac{1}{\det(A)} \)

Proof (in simple steps):

Step 1: We know, det(A × B) = det(A) × det(B)\(det(A × B) = det(A) × det(B)\)

 

Step 2: Also, \(A × A⁻¹ = I\) (by inverse matrix property)

 

Step 3: So, \(det(A × A⁻¹) = det(I)\)

\( \det(A^{-1}) = \frac{1}{\det(A)} \)

\( \det(A^{-1}) = \frac{1}{\det(A)} \)


Step 4: Thus, the determinant of the inverse is the reciprocal of the original determinant, as shown.

Understanding the inverse of a matrix is crucial for students in various fields. And here are some useful tips and tricks to master the inverse of a matrix. 

 

• Know the condition for inversibility: Only square matrices with a non-zero determinant have an inverse. Always check the determinant first (det(𝐴) ≠ 0) before attempting to find the inverse.
  
   
• Learn different methods: Familiarize yourself with various techniques such as using the adjoint formula, row reduction to the identity matrix, and formulas for 2 × 2 matrices. Different methods are useful in different contexts.
  
   
• Memorize the 2 × 2 Matrix Inverse Formula: For a 2 × 2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the inverse is: 
  
   
• Practice row operations for larger matrices: For 3 × 3 or larger matrices, use augmented matrices and row-reduction (Gaussian elimination) to find the inverse. Regular practice builds accuracy and speed.
  
   
• Always verify that \(𝐴⋅𝐴−1=𝐼\) (the identity matrix). This ensures that your inverse is correct and reinforces understanding of matrix multiplication.

The inverse of a matrix is not just a mathematical concept; it has numerous practical uses beyond math. Here are a few real-life instances where it can be applied.

• Inverse matrices play a crucial role in cryptography, particularly in classical encryption methods like the Hill cipher. To decrypt the message, the inverse of the key matrix is applied, effectively reversing the encryption process and ensuring secure communication.

• Inverse matrices provide an efficient method for solving systems of linear equations. By representing the system as a matrix equation \(Ax=b\), where 𝐴 is the coefficient matrix, x is the vector of variables, and 𝑏 is the constants vector, the solution can be found using \(x = A−1b\), assuming 𝐴 is invertible.

• In economics, input-output models analyze the relationships between different sectors of an economy.
  
  The Leontief inverse, derived from the inverse of a matrix, helps determine how changes in one sector affect others aiding in policymaking and economic planning.
  
  
   
• In computer graphics, transformations such as translation, rotation, and scaling are represented by matrices.
  
  The inverse of these transformation matrices allows for reversing or undoing transformations, which is essential in operations like object manipulation, camera movements, and rendering scenes from different perspectives.
  
  
   
• Control systems engineering employs inverse matrices to analyze and design systems that regulate variables like temperature, speed, and pressure. By using inverse matrices, engineers can model system behaviors, predict responses to inputs, and design controllers that ensure stability and desired performance.