Solving system of equations using matrices

**The inverse matrix method**

Systems with two equations and two avariables can also be solved using matrices and the inverse matrix.

First, arrange the system in the following form:

Next, create two matrices from the given system of equations:

1. The **coefficient matrix**, created by using the coefficients of the variables involved. So for our system, the coefficient matrix is:

**A=**
2. The **constant matrix**. It is created from the constants on the right side of the equal signs. In our system, the constant matrix is:

**B=**
This is equivalent to writing:

which is equivalent to the original two equations (check the multiplication yourself).

Represented using capital letters, we can say that: **A·X=B** (called **matrix equation**).

Then solve for the matrix variable X by left-multiplying both sides of the above matrix equation (**A·X=B**) by **A**^{-1}. That is:

**A·X=B** => **A**^{-1}·A·X=A^{-1}·B => X=A^{-1}·B·.

Use matrices to solve the system:

A=

X=

B=

To solve the system, we need the inverse of A. A

^{-1}=

So the solution of the system is given by

**X=A**^{-1}·B.

Then:

The solution is x=1, y=1