Find the Inverse [[4,10,30],[10,30,100],[30,100,354]]

Problem

[[4,10,30],[10,30,100],[30,100,354]](−1)

Solution

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

|A|=4*(30⋅354−100⋅100)−10*(10⋅354−30⋅100)+30*(10⋅100−30⋅30)

|A|=4*(10620−10000)−10*(3540−3000)+30*(1000−900)

|A|=4*(620)−10*(540)+30*(100)

|A|=2480−5400+3000=80

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)=620,(M_12)=540,(M_13)=100

(M_21)=10*(354)−30*(100)=540

(M_22)=4*(354)−30*(30)=516

(M_23)=4*(100)−10*(30)=100

(M_31)=10*(100)−30*(30)=100

(M_32)=4*(100)−10*(30)=100

(M_33)=4*(30)−10*(10)=20

3. Apply the cofactor signs using the pattern +−+ on each row to create the cofactor matrix C

C=[[620,−540,100],[−540,516,−100],[100,−100,20]]

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

adj(A)=[[620,−540,100],[−540,516,−100],[100,−100,20]]

5. Divide by the determinant to find the inverse matrix A(−1)=1/|A|*adj(A)

A(−1)=1/80*[[620,−540,100],[−540,516,−100],[100,−100,20]]

6. Simplify the fractions in the resulting matrix.

A(−1)=[[620/80,(−540)/80,100/80],[(−540)/80,516/80,(−100)/80],[100/80,(−100)/80,20/80]]

Final Answer

[[4,10,30],[10,30,100],[30,100,354]](−1)=[[7.75,−6.75,1.25],[−6.75,6.45,−1.25],[1.25,−1.25,0.25]]

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