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

Problem

[[−2,3,5],[6,−1,0],[4,2,1]](−1)

Solution

1. Calculate the determinant of the matrix A using cofactor expansion along the second row.

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

det(A)=−6*(3−10)−1*(−2−20)

det(A)=−6*(−7)−1*(−22)

det(A)=42+22=64

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)=6,(M_13)=16

(M_21)=−7,(M_22)=−22,(M_23)=−16

(M_31)=5,(M_32)=−30,(M_33)=−16

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

C=[[−1,−6,16],[7,−22,16],[5,30,−16]]

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

adj(A)=[[−1,7,5],[−6,−22,30],[16,16,−16]]

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

A(−1)=1/64*[[−1,7,5],[−6,−22,30],[16,16,−16]]

Final Answer

[[−2,3,5],[6,−1,0],[4,2,1]](−1)=[[−1/64,7/64,5/64],[−3/32,−11/32,15/32],[1/4,1/4,−1/4]]

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