Solve Using an Inverse Matrix 2x+3y=8 , 3x-y=1

Problem

{[2*x+3*y=8],[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=[[2,3],[3,−1]]

X=[[x],[y]]

B=[[8],[1]]

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

det(A)=(2)*(−1)−(3)*(3)

det(A)=−2−9=−11

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

A(−1)=1/(−11)*[[−1,−3],[−3,2]]

A(−1)=[[1/11,3/11],[3/11,−2/11]]

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

X=[[1/11,3/11],[3/11,−2/11]]*[[8],[1]]

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

x=(1/11)*(8)+(3/11)*(1)=8/11+3/11=11/11=1

y=(3/11)*(8)+(−2/11)*(1)=24/11−2/11=22/11=2

Final Answer

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

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