Find the Inverse [[5,13,57],[13,57,289],[57,289,1569]]

Problem

[[5,13,57],[13,57,289],[57,289,1569]](−1)

Solution

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

|A|=5*(57⋅1569−289⋅289)−13*(13⋅1569−57⋅289)+57*(13⋅289−57⋅57)

|A|=5*(89433−83521)−13*(20397−16473)+57*(3757−3249)

|A|=5*(5912)−13*(3924)+57*(508)

|A|=29560−51012+28956=7504

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)=5912,(M_12)=3924,(M_13)=508

(M_21)=13⋅1569−57⋅289=3924

(M_22)=5⋅1569−57⋅57=4596

(M_23)=5⋅289−13⋅57=704

(M_31)=13⋅289−57⋅57=508

(M_32)=5⋅289−13⋅57=704

(M_33)=5⋅57−13⋅13=116

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

C=[[5912,−3924,508],[−3924,4596,−704],[508,−704,116]]

4. Transpose the cofactor matrix to find the adjugate matrix adj(A) Since the matrix is symmetric, the transpose is identical.

adj(A)=[[5912,−3924,508],[−3924,4596,−704],[508,−704,116]]

5. Divide by the determinant |A|=7504 to find the inverse matrix A(−1)

A(−1)=1/7504*[[5912,−3924,508],[−3924,4596,−704],[508,−704,116]]

6. Simplify the fractions by dividing each term by the greatest common divisor.

5912/7504=739/938

(−3924)/7504=−981/1876

508/7504=127/1876

4596/7504=1149/1876

(−704)/7504=−11/117.25=−44/469

116/7504=29/1876

Final Answer

[[5,13,57],[13,57,289],[57,289,1569]](−1)=[[739/938,−981/1876,127/1876],[−981/1876,1149/1876,−44/469],[127/1876,−44/469,29/1876]]

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