Solve Using an Inverse Matrix 4r+7s=23 , r-2s=17

Problem

{[4*r+7*s=23],[r−2*s=17])

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=[[4,7],[1,−2]]

X=[[r],[s()]]

B=[[23],[17]]

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

det(A)=(4)*(−2)−(7)*(1)

det(A)=−8−7

det(A)=−15

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

A(−1)=1/(−15)*[[−2,−7],[−1,4]]

A(−1)=[[2/15,7/15],[1/15,−4/15]]

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

X=A(−1)*B

[[r],[s()]]=[[2/15,7/15],[1/15,−4/15]]*[[23],[17]]

5. Perform the matrix multiplication to find the values of r and s()

r=2/15*(23)+7/15*(17)=(46+119)/15=165/15=11

s=1/15*(23)−4/15*(17)=(23−68)/15=(−45)/15=−3

Final Answer

[[r],[s()]]=[[11],[−3]]

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