Find the Inverse

Problem

[[17/26,17/26,17/26],[17/26,12/26,17/26],[17/26,17/26,16/26]]

Solution

1. Factor out the common scalar 1/26 to simplify calculations, letting A=1/26*M where M is the integer matrix.

M=[[17,17,17],[17,12,17],[17,17,16]]

2. Calculate the determinant of M using cofactor expansion along the first row.

det(M)=17*(12⋅16−17⋅17)−17*(17⋅16−17⋅17)+17*(17⋅17−17⋅12)

det(M)=17*(192−289)−17*(272−289)+17*(289−204)

det(M)=17*(−97)−17*(−17)+17*(85)

det(M)=−1649+289+1445=85

3. Find the adjugate matrix adj(M) by calculating the transpose of the matrix of cofactors.

(C_11)=(12⋅16−17⋅17)=−97

(C_12)=−(17⋅16−17⋅17)=17

(C_13)=(17⋅17−17⋅12)=85

(C_21)=−(17⋅16−17⋅17)=17

(C_22)=(17⋅16−17⋅17)=0

(C_23)=−(17⋅17−17⋅17)=0

(C_31)=(17⋅17−17⋅12)=85

(C_32)=−(17⋅17−17⋅17)=0

(C_33)=(17⋅12−17⋅17)=−85

adj(M)=[[−97,17,85],[17,0,0],[85,0,−85]]

4. Apply the inverse formula A(−1)=(k*M)(−1)=1/k*M(−1)=26⋅1/det(M)*adj(M)

A(−1)=26/85*[[−97,17,85],[17,0,0],[85,0,−85]]

5. Distribute the scalar to find the final entries of the inverse matrix.

(26⋅(−97))/85=−2522/85

(26⋅17)/85=26/5=442/85

(26⋅85)/85=26

Final Answer

[[17/26,17/26,17/26],[17/26,12/26,17/26],[17/26,17/26,16/26]](−1)=[[−2522/85,26/5,26],[26/5,0,0],[26,0,−26]]

Want more problems? Check here!