Solve Using an Inverse Matrix -7x-2y=-13 , x-2y=11

Problem

{[−7*x−2*y=−13],[x−2*y=11])

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=[[−7,−2],[1,−2]]

X=[[x],[y]]

B=[[−13],[11]]

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

det(A)=(−7)*(−2)−(−2)*(1)

det(A)=14+2=16

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

A(−1)=1/16*[[−2,2],[−1,−7]]

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

X=A(−1)*B

X=1/16*[[−2,2],[−1,−7]]*[[−13],[11]]

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

X=1/16*[[(−2)*(−13)+(2)*(11)],[(−1)*(−13)+(−7)*(11)]]

X=1/16*[[26+22],[13−77]]

X=1/16*[[48],[−64]]

X=[[3],[−4]]

Final Answer

(x,y)=(3,−4)

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