Using Matrices To Solve A System Of Equations Or Simultaneous Equations

Simultaneous equations or system of equations of the form:
ax + by = h
cx + dy = k
can be solved using algebra.
Simultaneous equations can also be solved using matrices.

How to Represent the System of Equations as a Matrix Equation
A system of linear equations can be represented in matrix form as:
AX = B
Where:
A is the coefficient matrix, containing the coefficients of the variables in the equations.
X is the variable matrix (a column matrix), containing the variables you want to solve for.
B is the constant matrix (a column matrix), containing the constants on the right side of the equations.

How To Solve Matrix Equations

The following diagrams show how to solve a systems of equations using matrices. Scroll down the page for more examples and solutions.

 

Methods to Solve using Matrices
There are several methods to solve the matrix equation AX = B for X. In this lesson, we will solve using the inverse matrix. We have other lessons that show how to solve matrices using Gaussian Elimination and the Gauss-Jordan Method.

Using the Inverse Matrix:
If the coefficient matrix A is invertible (meaning its determinant is not zero), you can find the solution by multiplying both sides of the equation by the inverse of A (denoted as A⁻¹):

A⁻¹AX = A⁻¹B
IX = A⁻¹B (A⁻¹A = I, where I is the identity matrix)
X = A⁻¹B (IX = X, any matrix multiplied with the identity matrix will be unchanged)

So, to solve for X, you need to:
Find the inverse of matrix A (A⁻¹).
Multiply A⁻¹ by matrix B.

Review how to find the inverse of a matrix, if necessary.

Matrices & Systems of Equations

Example:
Using matrices, calculate the values of x and y for the following simultaneous equations:
2x – 2y – 3 = 0
8 y = 7x + 2

Solution:
Step 1: Write the equations in the form ax + by = c
2x – 2y – 3 = 0 ⇒ 2x – 2y = 3
8y = 7x + 2 ⇒ 7x – 8y = –2

Step 2: Write the equations in matrix form.

Step 3: Find the inverse of the 2 × 2 matrix.
Determinant = (2 × –8) – (–2 × 7) = – 2

Step 4: Multiply both sides of the matrix equations with the inverse.

So, x = 14 and y = 12.5

How Matrices Can Be Used To Solve A System Of Equations?

Using the inverse of a matrix to solve a system of equations.
3x + 2y = 7
-6x + 6y = 6

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How To Use A Matrix Equation To Solve A System Of Equations?

This video shows how to solve a system of equations by using a matrix equation.
AX = B
A^-1AX = A^-1B
IX = A^-1B
X = A^-1B

A 2 x 2 example and a 3 x 3 example are given.

Example:
Solve the system using a matrix equation
3x - y = 5
2x + y = 5

Example:
Solve the system using a matrix equation
x - 3y + 3z = -4
2x + 3y - z = 15
4x - 3y - z = 19

The graphing calculator is integrated into the lesson.

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How To Solve Simultaneous Equations Using Matrices?

Step by step solution.

Example:
2x + 3y = 8
x - 2y = -3

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Simultaneous Equations - Matrix Method

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