Gaussian Elimination and Cramer's Rule, we are going to use a different method. This method involves using 2 x 2 inverse matrices. To solve the linear system, we find the inverse of the 2 x 2 coefficient matrix (by using either row matrix operation or the formula) and multiply it with the answer column. Multiplying them would result in a column matrix, and the entries in the column matrix will give you a unique solution to the linear system.">

Solving linear systems using 2 x 2 inverse matrices

Solving linear systems using 2 x 2 inverse matrices

Now that we learned how to solve linear systems with Gaussian Elimination and Cramer's Rule, we are going to use a different method. This method involves using 2 x 2 inverse matrices. To solve the linear system, we find the inverse of the 2 x 2 coefficient matrix (by using either row matrix operation or the formula) and multiply it with the answer column. Multiplying them would result in a column matrix, and the entries in the column matrix will give you a unique solution to the linear system.

Lessons

Back then we learned that the linear system
$1x+2y=3$
$4x+5y=6$

Can be represented as the matrix

Now we can actually represent this in another way without the variables disappearing, which is

Now let , and . Then we can shorten the equation to be .

Now multiplying both sides of the equation by $A^{-1}$ will give us

We know that $A^{-1} A=I$, so then our equation becomes .

We also know that , and so our final equation is

With this equation, we can solve (which has the variable $x$ and $y$) simply by finding the inverse of $A$, and multiplying it by $b$.
• Introduction
Solving linear systems using inverse matrices overview

• 1.
Solving the system of equations using inverse matrices
You are given $A$ and $b$. Knowing that , solve the following linear systems by finding the inverse matrices and using the equation .
a)

b)

c)

d)

e)

f)