The structure shown below composed of metal bars was conside

The structure, shown below, composed of metal bars was considered in class. The temperature of the end of each bar is given in the figure. The temperatures x_1,x_2 and x_3 are the averages of the three adjacent temperatures. The system of equations, Ax =b, was shown in class to be: Use cofactors to find the inverse of the coefficient matrix A. Use A^-1 to solve for x.

Solution

Solution:

A = 3-1-1-13-1-1-13

Find matrix determinant :

det A = 16

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The determinant of is not zero, therefore the inverse matrix A^-1 exist. To calculate the inverse matrix find additional minors and cofactors of matrix

Find the minor M₁₁ and the cofactor C₁₁. In matrix A cross out row 1 and column 1.

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C₁₁ = (-1)¹⁺¹M₁₁ = 8

Find the minor M₁₂ and the cofactor C₁₂. In matrix A cross out row 1 and column 2.

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C₁₂ = (-1)¹⁺²M₁₂ = 4

Find the minor M₁₃ and the cofactor C₁₃. In matrix A cross out row 1 and column 3.

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C₁₃ = (-1)¹⁺³M₁₃ = 4

Find the minor M₂₁ and the cofactor C₂₁. In matrix A cross out row 2 and column 1.

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C₂₁ = (-1)²⁺¹M₂₁ = 4

Find the minor M₂₂ and the cofactor C₂₂. In matrix A cross out row 2 and column 2.

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C₂₂ = (-1)²⁺²M₂₂ = 8

Find the minor M₂₃ and the cofactor C₂₃. In matrix A cross out row 2 and column 3.

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C₂₃ = (-1)²⁺³M₂₃ = 4

Find the minor M₃₁ and the cofactor C₃₁. In matrix A cross out row 3 and column 1.

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C₃₁ = (-1)³⁺¹M₃₁ = 4

Find the minor M₃₂ and the cofactor C₃₂. In matrix A cross out row 3 and column 2.

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C₃₂ = (-1)³⁺²M₃₂ = 4

Find the minor M₃₃ and the cofactor C₃₃. In matrix A cross out row 3 and column 3.

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C₃₃ = (-1)³⁺³M₃₃ = 8

Write matrix of cofactors:

C = 844484448

Transposed matrix of cofactors:

C^T = 844484448

Find inverse matrix:

A^-1 = C^Tdet A = 121414141214141412

A-1= 1/2 1/4 1/4

1/4 1/2 1/4

1/4 1/4 1/2

B) x(1) = 1/2a + 1/4b+ 1/4c

x(2) = 1/4a + 1/2b + 1/4c

x(3) = 1/4a + 1/4b + 1/4c