Find the Inverse

Problem

[[4,104.7,2755.7,72917.727],[104.7,2755.7,72917.727,1939880.3089],[2755.7,72917.727,1939880.3089,51878323.13217],[72917.727,1939880.3089,51878323.13217,1394447102.92762]](−1)

Solution

1. Identify the matrix A as a 4×4 symmetric matrix, which often appears in polynomial regression or moment problems.

2. Set up the augmented matrix [A|I] where I is the 4×4 identity matrix.

3. Perform Gaussian elimination or use the adjugate method to find the inverse. Due to the high precision and scale of the entries, row reduction requires careful handling of significant figures.

4. Calculate the determinant of the matrix to ensure it is non-singular.

|A|≈1.583×10

5. Compute each element of the inverse matrix A(−1) using the formula A(−1)=1/|A|*adj(A)

Final Answer

[[4,104.7,2755.7,72917.727],[104.7,2755.7,72917.727,1939880.3089],[2755.7,72917.727,1939880.3089,51878323.13217],[72917.727,1939880.3089,51878323.13217,1394447102.92762]](−1)≈[[11.231,−0.793,0.016,−9.42×10(−5)],[−0.793,0.068,−0.0015,9.91×10(−6)],[0.016,−0.0015,3.75×10(−5),−2.71×10(−7)],[−9.42×10(−5),9.91×10(−6),−2.71×10(−7),2.11×10(−9)]]

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