Solve Using an Inverse Matrix 5x-4y-6z=-3 , x-3y+z=-1 , -3x-6y+7z=1

Problem

{[5*x−4*y−6*z=−3],[x−3*y+z=−1],[−3*x−6*y+7*z=1])

Solution

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

A=[[5,−4,−6],[1,−3,1],[−3,−6,7]]

X=[[x],[y],[z]]

B=[[−3],[−1],[1]]

2. Calculate the determinant of A denoted as |A| using the first row.

|A|=5*((−3)*(7)−(1)*(−6))−(−4)*((1)*(7)−(1)*(−3))+(−6)*((1)*(−6)−(−3)*(−3))

|A|=5*(−21+6)+4*(7+3)−6*(−6−9)

|A|=5*(−15)+4*(10)−6*(−15)

|A|=−75+40+90=55

3. Find the matrix of cofactors C by calculating the minor of each element.

(C_11)=−15,(C_12)=−10,(C_13)=−15

(C_21)=64,(C_22)=17,(C_23)=42

(C_31)=−22,(C_32)=−11,(C_33)=−11

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

adj(A)=[[−15,64,−22],[−10,17,−11],[−15,42,−11]]

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

A(−1)=1/55*[[−15,64,−22],[−10,17,−11],[−15,42,−11]]

6. Solve for X by multiplying A(−1) and B

X=A(−1)*B=1/55*[[−15,64,−22],[−10,17,−11],[−15,42,−11]]*[[−3],[−1],[1]]

x=1/55*((−15)*(−3)+(64)*(−1)+(−22)*(1))=(45−64−22)/55=(−41)/55

y=1/55*((−10)*(−3)+(17)*(−1)+(−11)*(1))=(30−17−11)/55=2/55

z=1/55*((−15)*(−3)+(42)*(−1)+(−11)*(1))=(45−42−11)/55=(−8)/55

Final Answer

[[x],[y],[z]]=[[−41/55],[2/55],[−8/55]]

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