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- Determinants and Inverses of Matrices
Solving linear systems using 2 x 2 inverse matrices
- Intro Lesson14:30
- Lesson: 1a5:37
- Lesson: 1b5:25
- Lesson: 1c5:04
- Lesson: 1d5:19
- Lesson: 1e7:47
- Lesson: 1f5:48
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−1A=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.
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



Now multiplying both sides of the equation by A−1 will give us

We know that A−1A=I, so then our equation becomes

We also know that


With this equation, we can solve

- IntroductionSolving 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)
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40.
Determinants and Inverses of Matrices
40.1
The determinant of a 2 x 2 matrix
40.2
The determinant of a 3 x 3 matrix (General & Shortcut Method)
40.3
The inverse of a 2 x 2 matrix
40.4
The inverse of 3 x 3 matrices with matrix row operations
40.5
The inverse of 3 x 3 matrix with determinants and adjugate
40.6
2 x 2 invertible matrix
40.7
Solving linear systems using Cramer's Rule
40.8
Solving linear systems using 2 x 2 inverse matrices