Find the Inverse [[1,2,3],[0,4,5],[1,0,6]]

Problem

[[1,2,3],[0,4,5],[1,0,6]](−1)

Solution

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

det(A)=1*(4⋅6−5⋅0)−0*(2⋅6−3⋅0)+1*(2⋅5−3⋅4)

det(A)=1*(24)+1*(10−12)

det(A)=24−2=22

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining after removing the row and column of each element.

(M_11)=24,(M_12)=−5,(M_13)=−4

(M_21)=12,(M_22)=3,(M_23)=−2

(M_31)=−2,(M_32)=5,(M_33)=4

3. Apply the cofactor signs using the pattern +−+ to create the matrix of cofactors C

C=[[24,5,−4],[−12,3,2],[−2,−5,4]]

4. Transpose the cofactor matrix to find the adjugate matrix adj(A)

adj(A)=[[24,−12,−2],[5,3,−5],[−4,2,4]]

5. Multiply by the reciprocal of the determinant to find the inverse matrix A(−1)=1/det(A)*adj(A)

A(−1)=1/22*[[24,−12,−2],[5,3,−5],[−4,2,4]]

Final Answer

[[1,2,3],[0,4,5],[1,0,6]](−1)=[[12/11,−6/11,−1/11],[5/22,3/22,−5/22],[−2/11,1/11,2/11]]

Want more problems? Check here!