Find the inverse of the (common) coefficient matrix of the two systems.

Question

Consider the following two systems

-6x-y=2

2x-9y=3

 

 

-6x-y=3

2x-9y=2

 

Find the inverse of the (common) coefficient matrix of the two systems.

A^-1=?

Answer 1

Use the fact that A z = B, need to find A^-1

 

Such that  A^-1 A z =  A^-1 B or z = A^-1  B

 

z is a 2 X 2 column matrix    x₁   x₂

                                          y₁   y₂

 

Both systems have the same coefficient matrix:

 

-6    -1

 2    -9

 

Call this matrix A and

 

2   3

3   2      is matrix B

 

A^-1    is   -.1607142857                .0178571492

              -.0357142857               -.1071428571

 

and A^-1 B  =   -.2678571429              -.4464285714

                       -.3928571429              -.3214285714

 

The solution of the first system is the first column of A^-1 B  and the solution to the second system is the second column of A^-1 B

 

The solution of the first system is x₁ = -.2678571429  and y₁ = -.3928571429

 

For the second system x₂ =  -.4464285714 and  y₂ =  -.3214285714