Solve Using an Inverse Matrix 2x+2y=2 , 3x-y=1

Problem

{[2*x+2*y=2],[3*x−y=1])

Solution

1. Write the system in matrix form A*X=B where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

A=[[2,2],[3,−1]]

X=[[x],[y]]

B=[[2],[1]]

2. Calculate the determinant of matrix A denoted as det(A) or |A|

|A|=(2)*(−1)−(2)*(3)

|A|=−2−6=−8

3. Find the inverse matrix A(−1) using the formula A(−1)=1/|A|*[[d,−b],[−c,a]]

A(−1)=1/(−8)*[[−1,−2],[−3,2]]

A(−1)=[[1/8,1/4],[3/8,−1/4]]

4. Solve for X by multiplying the inverse matrix by the constant matrix: X=A(−1)*B

X=[[1/8,1/4],[3/8,−1/4]]*[[2],[1]]

5. Perform the matrix multiplication to find the values of x and y

x=(1/8)*(2)+(1/4)*(1)=1/4+1/4=1/2

y=(3/8)*(2)+(−1/4)*(1)=3/4−1/4=1/2

Final Answer

[[x],[y]]=[[1/2],[1/2]]

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