Solve Using an Inverse Matrix 5x+4y=-2 , 3x-3y=2

Problem

{[5*x+4*y=−2],[3*x−3*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=[[5,4],[3,−3]]

X=[[x],[y]]

B=[[−2],[2]]

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

det(A)=(5)*(−3)−(4)*(3)

det(A)=−15−12=−27

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

A(−1)=1/(−27)*[[−3,−4],[−3,5]]

A(−1)=[[3/27,4/27],[3/27,−5/27]]

A(−1)=[[1/9,4/27],[1/9,−5/27]]

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

X=A(−1)*B

[[x],[y]]=[[1/9,4/27],[1/9,−5/27]]*[[−2],[2]]

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

x=(1/9)*(−2)+(4/27)*(2)=−2/9+8/27=−6/27+8/27=2/27

y=(1/9)*(−2)+(−5/27)*(2)=−2/9−10/27=−6/27−10/27=−16/27

Final Answer

(x,y)=(2/27,−16/27)

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