Solution of System of Linear Equations Using Inverse of a Matrix

4.6.1 Solution of System of Linear Equations Using Inverse of a Matrix

Matrix Representation of a Linear System

Consider a system of three linear equations in three variables $x$, $y$, $z$:

We can write this system compactly using matrices. Define:

• Coefficient matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$
• Variable matrix $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$
• Constant matrix $B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}$

Then the system becomes the single matrix equation:

Every system of linear equations can be written in this form when the number of equations equals the number of unknowns.

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Case I: Non-Singular Coefficient Matrix (Unique Solution)

When $A$ is a non-singular matrix, its determinant is non-zero ($|A| \neq 0$), and its inverse $A^{-1}$ exists.

Starting from $AX = B$, multiply both sides on the left by $A^{-1}$:

By the associative property of matrix multiplication:

Since $A^{-1}A = I$ (the identity matrix):

And since $IX = X$, we obtain:

This is the matrix method for solving a system of linear equations.

Procedure for the Matrix Method

1. Write the system in matrix form $AX = B$
2. Compute $|A|$. If $|A| \neq 0$, proceed
3. Find $A^{-1}$ using $A^{-1} = \frac{1}{|A|} \text{adj}(A)$
4. Compute $X = A^{-1}B$
5. Read off the values of $x$, $y$, $z$ from $X$

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Case II: Singular Coefficient Matrix (No Solution or Infinite Solutions)

When $A$ is a singular matrix, $|A| = 0$ and $A^{-1}$ does not exist. In this case, we examine the product $(\text{adj }A)B$.

Subcase 1: $(\text{adj }A)B \neq O$ (zero matrix)

The system has no solution. Such a system is called inconsistent. …