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
Notes:
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 $b$.

2.
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 .