Find the Inverse [[4,-10,29],[1,-2,5],[-3,7,-19]]

Problem

[[4,−10,29],[1,−2,5],[−3,7,−19]]

Solution

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

det(A)=4*((−2)*(−19)−(5)*(7))−(−10)*((1)*(−19)−(5)*(−3))+29*((1)*(7)−(−2)*(−3))

det(A)=4*(38−35)+10*(−19+15)+29*(7−6)

det(A)=4*(3)+10*(−4)+29*(1)

det(A)=12−40+29=1

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)=3,(M_12)=−4,(M_13)=1

(M_21)=190−203=−13,(M_22)=−76−(−87)=11,(M_23)=28−30=−2

(M_31)=−50−(−58)=8,(M_32)=20−29=−9,(M_33)=−8−(−10)=2

3. Apply the cofactor signs using the pattern +−+ on the matrix of minors.

C=[[3,4,1],[13,11,2],[8,9,2]]

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

adj(A)=[[3,13,8],[4,11,9],[1,2,2]]

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

A(−1)=1/1*[[3,13,8],[4,11,9],[1,2,2]]

Final Answer

[[4,−10,29],[1,−2,5],[−3,7,−19]](−1)=[[3,13,8],[4,11,9],[1,2,2]]

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