Solve Using an Inverse Matrix x+y=-2 , x-2y=10

Problem

{[x+y=−2],[x−2*y=10])

Solution

1. Write the system as a matrix equation of the form A*X=B

A=[[1,1],[1,−2]]

X=[[x],[y]]

B=[[−2],[10]]

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

det(A)=(1)*(−2)−(1)*(1)

det(A)=−2−1=−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],[−1,1]]

A(−1)=[[2/3,1/3],[1/3,−1/3]]

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

X=[[2/3,1/3],[1/3,−1/3]]*[[−2],[10]]

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

x=(2/3)*(−2)+(1/3)*(10)=−4/3+10/3=6/3=2

y=(1/3)*(−2)+(−1/3)*(10)=−2/3−10/3=−12/3=−4

Final Answer

[[x],[y]]=[[2],[−4]]

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