Solve Using an Inverse Matrix 2x+y=8 , x+2y=1

Problem

{[2*x+y=8],[x+2*y=1])

Solution

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

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

X=[[x],[y]]

B=[[8],[1]]

2. Calculate the determinant of matrix A denoted as det(A) using the formula a*d−b*c

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 vector B such that X=A(−1)*B

X=1/3*[[2,−1],[−1,2]]*[[8],[1]]

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

X=1/3*[[(2)*(8)+(−1)*(1)],[(−1)*(8)+(2)*(1)]]

X=1/3*[[15],[−6]]

X=[[5],[−2]]

Final Answer

[[x],[y]]=[[5],[−2]]

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