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.
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 1x+2y=3
4x+5y=6 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

Now let let A and x = the matrix, and let b=matrix. Then we can shorten the equation to be ax=b.

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

We know that A−1A=IA^{-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 xx and yy) simply by finding the inverse of AA, and multiplying it by bb.
  • Introduction
    Solving linear systems using inverse matrices overview
  • 1.
    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.
    a)
    Solving linear systems using 2 x 2 inverse matrices

    b)
    Solving linear systems using 2 x 2 inverse matrices

    c)
    Solving linear systems using 2 x 2 inverse matrices

    d)
    Solving linear systems using 2 x 2 inverse matrices

    e)
    Solving linear systems using 2 x 2 inverse matrices

    f)
    Solving linear systems using 2 x 2 inverse matrices

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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
AU Year 10 Maths