Solve Using an Inverse Matrix x+y=4 , x-y=2

Problem

{[x+y=4],[x−y=2])

Solution

1. Write the system of linear equations in the matrix form A*X=B

A=[[1,1],[1,−1]]

X=[[x],[y]]

B=[[4],[2]]

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

det(A)=(1)*(−1)−(1)*(1)

det(A)=−1−1=−2

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

A(−1)=1/(−2)*[[−1,−1],[−1,1]]

A(−1)=[[0.5,0.5],[0.5,−0.5]]

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

X=[[0.5,0.5],[0.5,−0.5]]*[[4],[2]]

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

x=(0.5)*(4)+(0.5)*(2)=2+1=3

y=(0.5)*(4)+(−0.5)*(2)=2−1=1

Final Answer

[[x],[y]]=[[3],[1]]

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