Solve Using an Inverse Matrix x+y+z=8 , 2x-y-z=10 , x-2y-3z=22

Problem

{[x+y+z=8],[2*x−y−z=10],[x−2*y−3*z=22])

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=[[1,1,1],[2,−1,−1],[1,−2,−3]]

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

B=[[8],[10],[22]]

2. Calculate the determinant of matrix A to ensure it is invertible.

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

det(A)=1*(3−2)−1*(−6+1)+1*(−4+1)

det(A)=1+5−3=3

3. Find the matrix of cofactors C for matrix A

(C_11)=1,(C_12)=5,(C_13)=−3

(C_21)=1,(C_22)=−4,(C_23)=3

(C_31)=0,(C_32)=3,(C_33)=−3

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

adj(A)=[[1,1,0],[5,−4,3],[−3,3,−3]]

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

A(−1)=1/3*[[1,1,0],[5,−4,3],[−3,3,−3]]

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

X=1/3*[[1,1,0],[5,−4,3],[−3,3,−3]]*[[8],[10],[22]]

X=1/3*[[(1)*(8)+(1)*(10)+(0)*(22)],[(5)*(8)+(−4)*(10)+(3)*(22)],[(−3)*(8)+(3)*(10)+(−3)*(22)]]

X=1/3*[[18],[66],[−60]]

X=[[6],[22],[−20]]

Final Answer

[[x],[y],[z]]=[[6],[22],[−20]]

Want more problems? Check here!