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

Problem

{[3*x−y−z=3],[−2*x+3*y+z=3],[4*x−2*y−2*z=2])

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=[[3,−1,−1],[−2,3,1],[4,−2,−2]]

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

B=[[3],[3],[2]]

2. Calculate the determinant of matrix A using the first row.

det(A)=3*(3*(−2)−(1)*(−2))−(−1)*((−2)*(−2)−(1)*(4))−1*((−2)*(−2)−(3)*(4))

det(A)=3*(−6+2)+1*(4−4)−1*(4−12)

det(A)=3*(−4)+1*(0)−1*(−8)

det(A)=−12+8=−4

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

(C_11)=−4,(C_12)=0,(C_13)=−8

(C_21)=0,(C_22)=−2,(C_23)=2

(C_31)=2,(C_32)=−1,(C_33)=7

C=[[−4,0,−8],[0,−2,2],[2,−1,7]]

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

adj(A)=CT=[[−4,0,2],[0,−2,−1],[−8,2,7]]

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

A(−1)=−1/4*[[−4,0,2],[0,−2,−1],[−8,2,7]]=[[1,0,−0.5],[0,0.5,0.25],[2,−0.5,−1.75]]

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

X=[[1,0,−0.5],[0,0.5,0.25],[2,−0.5,−1.75]]*[[3],[3],[2]]

x=(1)*(3)+(0)*(3)+(−0.5)*(2)=3−1=2

y=(0)*(3)+(0.5)*(3)+(0.25)*(2)=1.5+0.5=2

z=(2)*(3)+(−0.5)*(3)+(−1.75)*(2)=6−1.5−3.5=1

Final Answer

[[x],[y],[z]]=[[2],[2],[1]]

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