Solving linear systems using 2 x 2 inverse matrices

Intro Lesson
14:30
Lesson: 1a
5:37
Lesson: 1b
5:25
Lesson: 1c
5:04
Lesson: 1d
5:19
Lesson: 1e
7:47
Lesson: 1f
5:48

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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.

Basic concepts: Notation of matrices, Matrix multiplication, The inverse of a 2 x 2 matrix,

Lessons

Back then we learned that the linear system

1x+2y=3

4x+5y=6

Can be represented as the matrix
linear system represented as a matrix

Now we can actually represent this in another way without the variables disappearing, which is
linear system represented as a matrix with variables

linear system represented as a matrix with variables

Now let

, and

. Then we can shorten the equation to be ax=b

.

Now multiplying both sides of the equation by

A^{-1}

will give us $multiplying the equation by A^{-1}$

We know that

A^{-1} A=I

, so then our equation becomes $equation: Ix=A^{-1}b$ .

We also know that equation: Ix=x

, and so our final equation is
$equation: Ix=A^{-1}b$
With this equation, we can solve overrightarrow{x}

(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)

Solving linear systems using 2 x 2 inverse matrices

Solving linear systems using 2 x 2 inverse matrices

Lessons

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