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

Problem

{[2*x+y=−2],[x+2*y=2])

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=[[2,1],[1,2]]

X=[[x],[y]]

B=[[−2],[2]]

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

det(A)=(2)*(2)−(1)*(1)

det(A)=3

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

A(−1)=1/3*[[2,−1],[−1,2]]

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

X=A(−1)*B

X=1/3*[[2,−1],[−1,2]]*[[−2],[2]]

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

X=1/3*[[(2)*(−2)+(−1)*(2)],[(−1)*(−2)+(2)*(2)]]

X=1/3*[[−6],[6]]

X=[[−2],[2]]

Final Answer

(x,y)=(−2,2)

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