Using Matrices to Solve a System of Equations or Simultaneous Equations

Simultaneous equations of the form ax + by = h and cx + dy = k can be solved using algebra.

Simultaneous equations can also be solved using matrices.

First, we would look at how the inverse of a matrix can be used to solve a matrix equation.

Given the matrix equation AY = B, find the matrix Y.

If we multiply each side of the equation by A^-1, we get

A^-1A Y = A^-1B

I Y = A ^-1B    ( AA ^-1= I, where I is the identity matrix)

Y = A ^-1B       ( IY = Y, any matrix multiply with the identity matrix will be unchanged)

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

The following video shows how matrices can be used to solve a system of equations.

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