Find the Inverse of the Resulting Matrix

Problem

[[r,1,1],[0,r,2*r],[−1,−1,−1]]*[[1,0,0],[0,1,0],[0,0,1]]

Solution

1. Identify the resulting matrix. Multiplying any matrix A by the identity matrix I results in the matrix A itself.

A=[[r,1,1],[0,r,2*r],[−1,−1,−1]]

2. Calculate the determinant of A using cofactor expansion along the first column.

det(A)=r*(r*(−1)−(2*r)*(−1))−0+(−1)*(1*(2*r)−1*(r))

det(A)=r*(−r+2*r)−(2*r−r)

det(A)=r(r)−r

det(A)=r^2−r

3. Find the matrix of cofactors C

(C_11)=r*(−1)−(2*r)*(−1)=r

(C_12)=−(0*(−1)−(2*r)*(−1))=−2*r

(C_13)=0*(−1)−r*(−1)=r

(C_21)=−(1*(−1)−1*(−1))=0

(C_22)=r*(−1)−1*(−1)=−r+1

(C_23)=−(r*(−1)−1*(−1))=r−1

(C_31)=1*(2*r)−1*(r)=r

(C_32)=−(r*(2*r)−1*(0))=−2*r^2

(C_33)=r(r)−1*(0)=r^2

4. Form the adjugate matrix adj(A) by transposing the cofactor matrix.

adj(A)=[[r,0,r],[−2*r,1−r,−2*r^2],[r,r−1,r^2]]

5. Apply the inverse formula A^(−1)=1/det(A)*adj(A)

A^(−1)=1/(r^2−r)*[[r,0,r],[−2*r,1−r,−2*r^2],[r,r−1,r^2]]

6. Simplify the entries by dividing each term by r*(r−1)

r/(r*(r−1))=1/(r−1)

(−2*r)/(r*(r−1))=(−2)/(r−1)

(1−r)/(r*(r−1))=(−(r−1))/(r*(r−1))=−1/r

(−2*r^2)/(r*(r−1))=(−2*r)/(r−1)

(r−1)/(r*(r−1))=1/r

(r^2)/(r*(r−1))=r/(r−1)

Final Answer

[[r,1,1],[0,r,2*r],[−1,−1,−1]]^(−1)=[[1/(r−1),0,1/(r−1)],[(−2)/(r−1),−1/r,(−2*r)/(r−1)],[1/(r−1),1/r,r/(r−1)]]

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