Using Inverse Matrix To Solve System Of Equations

Using Inverse Matrix to Solve System of Equations

Linear systems of equations are fundamental in mathematics, engineering, and applied sciences. One powerful method to tackle these systems is by using the inverse matrix approach. But this technique leverages the properties of matrix algebra to find solutions efficiently, especially for square systems where the number of equations matches the number of unknowns. When dealing with multiple variables, solving such systems manually can become cumbersome. In this article, we’ll explore how the inverse matrix method works, its mathematical foundation, and practical applications.

Steps to Solve a System Using the Inverse Matrix

To solve a system of linear equations using the inverse matrix, follow these steps:

1. Express the System in Matrix Form:
   Convert the system of equations into the matrix equation Ax = b, where:

   • A is the coefficient matrix,
   • x is the column vector of variables,
   • b is the column vector of constants.

   To give you an idea, the system:
   $
   \begin{cases}
   2x + 3y = 5 \
   4x + y = 6
   \end{cases}
   $
   becomes:
   $
   \begin{bmatrix}
   2 & 3 \
   4 & 1
   \end{bmatrix}
   \begin{bmatrix}
   x \
   y
   \end{bmatrix}

   \begin{bmatrix}
   5 \
   6
   \end{bmatrix}.
   $

2. Check if the Matrix A is Invertible:
   A matrix has an inverse only if its determinant is non-zero. For a 2×2 matrix:
   $
   \text{det}(A) = ad - bc.
   $
   If the determinant is zero, the system either has no solution or infinitely many solutions It's one of those things that adds up..

3. Find the Inverse of Matrix A:
   For a 2×2 matrix:
   $
   A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A),
   $
   where adj(A) is the adjugate matrix (transpose of the cofactor matrix). For larger matrices, methods like Gaussian elimination or the adjugate formula are used.

4. Multiply A⁻¹ by b:
   Once A⁻¹ is found, compute:
   $
   x = A^{-1}b.
   $
   This gives the solution vector directly.

Example: Solving a 2×2 System

Let’s apply the inverse matrix method to the system:
$
\begin{cases}
2x + 3y = 5 \
4x + y = 6
\end{cases}.
$

Step 1: Matrix Form
$
A = \begin{bmatrix} 2 & 3 \ 4 & 1 \end{bmatrix}, \quad x = \begin{bmatrix} x \ y \end{bmatrix}, \quad b = \begin{bmatrix} 5 \ 6 \end{bmatrix}.
$

Step 2: Determinant of A
$
\text{det}(A) = (2)(1) - (3)(4) = 2 - 12 = -10 \neq 0.
$
Since the determinant is non-zero, A⁻¹ exists.

Step 3: Find A⁻¹
For a 2×2 matrix:
$
A^{-1} = \frac{1}{-10} \cdot \begin{bmatrix} 1 & -3 \ -4 & 2 \end{bmatrix} = \begin{bmatrix} -0.1 & 0.3 \ 0.4 & -0.2 \end{bmatrix}.
$

Step 4: Compute x = A⁻¹b
$
x = \begin{bmatrix} -0.1 & 0.3 \ 0.4 & -0.2 \end{bmatrix} \begin{bmatrix} 5 \ 6 \end{bmatrix} = \begin{bmatrix} (-0.1)(5) + (0.3)(6) \ (0.4)(5) + (-0.2)(6) \end{bmatrix} = \begin{bmatrix} 1.3 \ 0.8 \end{bmatrix}.
$
Thus, the solution is x = 1.3 and y = 0.8.

Scientific Explanation: Why the Inverse Matrix Works

The inverse matrix method is rooted in linear algebra principles. When a system is expressed as Ax = b, multiplying both sides by A⁻¹ gives:
$
A^{-1}Ax = A^{-1}b \implies Ix = A^{-1}b \implies x = A^{-1}b.
$
Here, I is the identity