Solve Using an Inverse Matrix e+3f-2g=11 , 2e-f+g=3 , 3e-2f+4g=1

Problem

{[e+3*ƒ−2*g=11],[2*e−ƒ+g=3],[3*e−2*ƒ+4*g=1])

Solution

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

A=[[1,3,−2],[2,−1,1],[3,−2,4]]

X=[[e],[ƒ],[g]]

B=[[11],[3],[1]]

2. Calculate the determinant of A denoted as |A| to ensure the matrix is invertible.

|A|=1*((−1)*(4)−(1)*(−2))−3*((2)*(4)−(1)*(3))+(−2)*((2)*(−2)−(−1)*(3))

|A|=1*(−2)−3*(5)−2*(−1)

|A|=−2−15+2=−15

3. Find the matrix of cofactors C for matrix A

C=[[−2,−5,−1],[−8,10,11],[1,−5,−7]]

4. Determine the adjugate matrix adj(A) by taking the transpose of the cofactor matrix CT

adj(A)=[[−2,−8,1],[−5,10,−5],[−1,11,−7]]

5. Compute the inverse matrix A(−1) using the formula A(−1)=1/|A|*adj(A)

A(−1)=1/(−15)*[[−2,−8,1],[−5,10,−5],[−1,11,−7]]

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

X=−1/15*[[(−2)*(11)+(−8)*(3)+(1)*(1)],[(−5)*(11)+(10)*(3)+(−5)*(1)],[(−1)*(11)+(11)*(3)+(−7)*(1)]]

X=−1/15*[[−45],[−30],[15]]

X=[[3],[2],[−1]]

Final Answer

[[e],[ƒ],[g]]=[[3],[2],[−1]]

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