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

Problem

[[0,1,−3],[2,3,−1],[4,5,−2]]

Solution

1. Calculate the determinant of the matrix A using the first row.

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

det(A)=0−1*(−4+4)−3*(10−12)

det(A)=0−0+6=6

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)=−1,(M_12)=0,(M_13)=−2

(M_21)=13,(M_22)=12,(M_23)=−4

(M_31)=8,(M_32)=6,(M_33)=−2

3. Apply the cofactor signs using the pattern of alternating signs to create the cofactor matrix C

C=[[−1,0,−2],[−13,12,4],[8,−6,−2]]

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

adj(A)=[[−1,−13,8],[0,12,−6],[−2,4,−2]]

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

A(−1)=1/6*[[−1,−13,8],[0,12,−6],[−2,4,−2]]

Final Answer

[[0,1,−3],[2,3,−1],[4,5,−2]](−1)=[[−1/6,−13/6,4/3],[0,2,−1],[−1/3,2/3,−1/3]]

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