Solving Linear Systems with Matrices.

 Solving linear systems is one of the fundamental concepts in linear algebra, widely applicable in engineering, physics, computer science, economics, and various other disciplines. Matrices offer a powerful and efficient method for handling these systems, especially when dealing with multiple variables and equations. This post will cover how to solve linear systems using matrices, with detailed examples and problem sets to help you gain practical understanding. We’ll explore different methods such as inverse matrices, Gaussian elimination, and Cramer’s Rule to solve systems of linear equations. Each method has its own advantages depending on the structure of the problem. Let’s get started!

illustration showing matrix methods like Gaussian elimination, inverses, and Cramer's rule for solving systems.

What is a Linear System?

A system of linear equations is a set of equations where each equation is linear, meaning each term is either a constant or the product of a constant and a variable. For example:

$$x+2y=4\text{and}3x+4y=10$$

These are linear equations because each variable is raised to the power of 1. A linear system typically has multiple equations and multiple unknowns.

The matrix form of a system of equations can be written as:

$$AX=B$$

Where:

• $A$ is the coefficient matrix.
• $X$ is the column vector of variables.
• $B$ is the column vector of constants.

Once in this form, matrix operations can be used to solve the system efficiently.

Solving Linear Systems Using Inverse Matrices

The inverse matrix method is one of the simplest ways to solve a system, but it only works if the coefficient matrix $A$ is square (same number of rows and columns) and invertible. The inverse matrix of $A$, denoted as $A^{-1}$, can be used to find the solution:

$$X=A^{-1}B$$

Here, $X$ is the vector of unknowns, $A^{-1}$ is the inverse of the coefficient matrix, and $B$ is the vector of constants.

Example 1: Consider the following system:

$$\begin{gathered} x+2y=5 \\ 3x+4y=6 \end{gathered}$$

In matrix form, it becomes:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}5\\ 6\end{array}\right]$$

By calculating $A^{-1}$, we can solve for $X$ and find the values of $x$ and $y$.

Example 2: For the system:

$$\begin{gathered} 2x+3y+z=9 \\ x+2y-z=3 \\ 3x-y+4z=8 \end{gathered}$$

The matrix form is:

$$A = \begin{bmatrix} 2 & 3 & 1 \\ 1 & 2 & -1 \\ 3 & -1 & 4 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 9 \\ 3 \\ 8 \end{bmatrix}$$

Using $A^{-1}$, we can find the values of $x$, $y$, and $z$.

Gaussian Elimination

Gaussian elimination is another popular method to solve systems of linear equations. It transforms the system into row echelon form using elementary row operations, making it easier to solve the equations step by step. Once in row echelon form, back substitution is used to solve for the variables.

The process involves three main steps:

1. Pivoting: Make the diagonal elements of the matrix 1.
2. Elimination: Eliminate the variables below the pivot by subtracting multiples of the row.
3. Back Substitution: Once the system is in upper triangular form, solve for the variables starting from the last row.

Example 1: Consider the system:

$$\begin{gathered} x+2y+z=4 \\ 2x+3y+2z=8 \\ x+y+2z=5 \end{gathered}$$

The augmented matrix is:

$$\begin{bmatrix} 1 & 2 & 1 & 4 \\ 2 & 3 & 2 & 8 \\ 1 & 1 & 2 & 5 \end{bmatrix}$$

Using Gaussian elimination, reduce the matrix to row echelon form and solve for $x$, $y$, and $z$.

Example 2: For the system:

$$\begin{gathered} x+y+z=6 \\ 2x+3y+z=11 \\ 3x+2y+2z=10 \end{gathered}$$

The augmented matrix is:

$$\begin{bmatrix} 1 & 1 & 1 & 6 \\ 2 & 3 & 1 & 11 \\ 3 & 2 & 2 & 10 \end{bmatrix}$$

Use Gaussian elimination to solve the system step by step, reducing it to row echelon form and solving for $x$, $y$, and $z$.

Cramer’s Rule

Cramer’s Rule is a method that uses determinants to solve a system of linear equations. It is applicable when the system has the same number of equations as unknowns, and the determinant of the coefficient matrix is non-zero. The solution for each variable is given by:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

Where $A_i$ is the matrix obtained by replacing the $i$-th column of $A$ with the column vector $B$, and $\det(A)$ is the determinant of matrix $A$.

Example 1: Solve the system using Cramer’s Rule:

$$\begin{gathered} x+2y=3 \\ 3x+4y=7 \end{gathered}$$

The coefficient matrix is:

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

Find the determinants $\det(A)$ and $\det(A_1)$, $\det(A_2)$ to solve for $x$ and $y$.

Example 2: For the system:

$$2x + y = 4 \\ x - y = 1$$

The matrix form is:

$$A = \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}$$

By using Cramer’s Rule, calculate the solution for $x$ and $y$.

Problem Set

Here are 20 problems for you to practice solving linear systems using matrices:

1. Solve the following system using inverse matrices:

   $$\begin{gathered} x+2y=5 \\ 3x+4y=6 \end{gathered}$$

2. Use Gaussian elimination to solve:

   $$\begin{gathered} 2x+3y=8 \\ 5x+y=7 \end{gathered}$$

3. Apply Cramer’s Rule to find the solution:

   $$\begin{gathered} 2x+y=6 \\ 4x+3y=10 \end{gathered}$$

4. Find the solution using inverse matrices:

   $$\begin{gathered} x-2y=4 \\ 3x+y=5 \end{gathered}$$

5. Solve this system using Gaussian elimination:

   $$\begin{gathered} x+2y+z=10 \\ 3x+4y+2z=20 \\ 2x+y+3z=14 \end{gathered}$$

6. Use Cramer’s Rule to solve:

   $$\begin{gathered} x+2y=3 \\ 3x+5y=7 \end{gathered}$$

7. Solve using Gaussian elimination:

   $$\begin{gathered} 4x+y=9 \\ 2x-y=3 \end{gathered}$$

8. Find the solution using inverse matrices:

   $$\begin{gathered} 2x+y=5 \\ 3x+4y=6 \end{gathered}$$

9. Apply Gaussian elimination to solve:

   $$\begin{gathered} x+y+z=4 \\ 2x+4y+z=10 \\ 3x+2y+4z=14 \end{gathered}$$

10. Use Cramer’s Rule to find the solution:

    $$\begin{gathered} x+3y=7 \\ 2x+5y=11 \end{gathered}$$

11. Solve using Gaussian elimination:

    $$\begin{gathered} 3x+2y=7 \\ 5x+4y=13 \end{gathered}$$

12. Use Cramer’s Rule to find the solution:

    $$\begin{gathered} x-2y=6 \\ 4x+y=12 \end{gathered}$$

13. Solve using inverse matrices:

    $$2x + 3y + z = 9 \\ x + 2y - z = 3 \\$$

14. Solve using inverse matrices:

    $$\begin{gathered} 4x+y+2z=15 \\ 3x+2y+z=11 \\ 5x+3y+4z=19 \end{gathered}$$

15. Apply Cramer’s Rule to find the solution:

    $$\begin{gathered} x+y+z=6 \\ 2x+3y+4z=15 \\ 4x+y+2z=14 \end{gathered}$$

16. Solve using Gaussian elimination:

    $$3x + 2y = 8 \\ 5x + y = 9$$

17. Use inverse matrices to solve:

    $$\begin{gathered} 2x+4y+z=13 \\ x+y+z=7 \\ 4x+5y+2z=19 \end{gathered}$$

18. Apply Gaussian elimination:

    $$\begin{gathered} 2x+y=6 \\ 4x+3y=14 \end{gathered}$$

19. Solve using Cramer’s Rule:

    $$x + 3y = 7 \\ 2x + 5y = 13$$

20. Find the solution using Gaussian elimination:

    $$\begin{gathered} x+y+z=10 \\ 2x+3y+z=16 \\ x+2y+2z=12 \end{gathered}$$