Matrix Inversion — Solving Linear Systems

 

Introduction

Matrix inversion is one of the most elegant tools in linear algebra. It allows for solving linear systems in a compact and efficient way. In this post, you will learn:

            🔹 What is an inverse matrix?
            🔹 How to invert a 2×2 matrix
            🔹 When is a matrix singular?
            🔹 How to use the inverse to solve AX = B
            🔹 How Cramer’s Rule works for a 2×2 matrix

   1.   What Is an Inverse Matrix?

We say that matrix A has an inverse when there exists a matrix A⁻¹ such that:

                                          AA⁻¹ = A⁻¹A = I,

where I is the identity matrix. Matrix A must be square and non-zero.

2. How to Invert a 2×2 Matrix

Example:

Steps:

     1: Define A⁻¹ — assume a matrix such that AA⁻¹ = I

              Therefore:

     2: Multiply the matrices — To find the inverse, we multiply the two matrices and – for the equation to be true – we need to solve the four equations:

     3: Solve the two systems — resulting in values for x, y, w, z                                                                                                                        

                                                                          

          The inverse of A is

Alternative method: for A⁻¹ of a 2×2 matrix, use the formula:

                                             

3. When Is a Matrix Singular?

A matrix is singular when it has no inverse. This happens when:

           🔹 The matrix is not square, or
           🔹 Its determinant is zero.

4. Using the Inverse to Solve AX = B

If A is invertible, we can isolate X:

                                                      AX = B ⇒ X = A⁻¹B.
Example:                                 

det(A)= 1x4-2x3=-2

5. Cramer’s Rule for 2×2 Systems

Cramer’s Rule offers a direct method for solving small systems (2x2, 3x3, even 4x4), using only determinants. Given the system:

a₁₁x + a₁₂y = b₁
a₂₁x + a₂₂y = b₂

Steps:

     1. Calculate D (determinant of coefficient matrix)                                                 

     2. Calculate Dx (replace x-column with constants)                                                                  

                                                      

🔹 If D ≠ 0 → unique solution
🔹 If D = 0 and Dx = Dy = 0 → infinitely many solutions
🔹 If D = 0 and either Dx ≠ 0 or Dy ≠ 0 → no solution

Example:

      

6. Why Learn Matrix Inversion?

Matrix inversion and Cramer’s Rule are powerful tools to solve linear systems. Matrices and determinants are essential in many fields. A central problem in linear algebra is solving AX = B. Even though it can be solved by other methods, it’s common to solve it using the inverse:

X = A⁻¹B.

Technically, we don’t divide matrices. The operation equivalent to 'dividing' matrix B by A is multiplying B by A⁻¹. Just like dividing 10 by 2 is the same as multiplying by ½:

10 ÷ 2 = 10 × ½ = 5

However, matrix multiplication does not follow the same rules as scalar arithmetic — in particular,

                                                               A × B⁻¹ ≠ B⁻¹ × A.

TIP

Learn the manual procedure for inversion, but later, use your preferred software. For beginners, many online tools are available.

Search for: INVERSE MATRIX CALCULATOR

🔗 https://www.omnicalculator.com/math/matrix-inverse

🔗 https://www.mathsisfun.com/algebra/matrix-inverse.html