Solve Using an Inverse Matrix y+z=-1 , -x+y+z=0 , x-z=-1

Problem

{[y+z=−1],[−x+y+z=0],[x−z=−1])

Solution

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

A=[[0,1,1],[−1,1,1],[1,0,−1]]

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

B=[[−1],[0],[−1]]

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

det(A)=0*(1*(−1)−0*(1))−1*((−1)*(−1)−1*(1))+1*((−1)*(0)−1*(1))

det(A)=0−1*(1−1)+1*(−1)

det(A)=−1

3. Find the adjugate matrix adj(A) by calculating the transpose of the cofactor matrix.

adj(A)=[[−1,1,0],[0,−1,−1],[−1,1,1]]

4. Determine the inverse matrix A^(−1) using the formula A^(−1)=1/det(A)*adj(A)

A^(−1)=1/(−1)*[[−1,1,0],[0,−1,−1],[−1,1,1]]

A^(−1)=[[1,−1,0],[0,1,1],[1,−1,−1]]

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

X=[[1,−1,0],[0,1,1],[1,−1,−1]]*[[−1],[0],[−1]]

x=(1)*(−1)+(−1)*(0)+(0)*(−1)=−1

y=(0)*(−1)+(1)*(0)+(1)*(−1)=−1

z=(1)*(−1)+(−1)*(0)+(−1)*(−1)=0

Final Answer

[[x],[y],[z]]=[[−1],[−1],[0]]

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