Solving linear systems using 2 x 2 inverse matrices - 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 1x+2y=3
4x+5y=6 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 A1A^{-1} will give us

We know that A1A=IA^{-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 xx and yy) simply by finding the inverse of AA, and multiplying it by bb.
  • 2.
    Solving the system of equations using inverse matrices
    You are given AA and bb. Knowing that Solving linear systems using 2 x 2 inverse matrices, solve the following linear systems by finding the inverse matrices and using the equation Solving linear systems using 2 x 2 inverse matrices.
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Solving linear systems using 2 x 2 inverse matrices

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