Hilbert Matrix/Examples/3x3/Determinant

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Examples of Hilbert Matrices

The determinant of the order $3$ Hilbert matrix is given by:

$\map \det {H_3} = \dfrac 1 {2160}$


Proof

\(\ds \map \det {H_3}\) \(=\) \(\ds \begin {vmatrix}

1 & \tfrac 1 2 & \tfrac 1 3 \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 \\ \end {vmatrix}\)

Definition of Hilbert Matrix
\(\ds \) \(=\) \(\ds 1 \times \begin {vmatrix} \dfrac 1 3 & \dfrac 1 4 \\ \dfrac 1 4 & \dfrac 1 5 \end {vmatrix} 
- \dfrac 1 2 \times \begin {vmatrix} \dfrac 1 2 & \dfrac 1 4 \\ \dfrac 1 3 & \dfrac 1 5 \end {vmatrix}
+ \dfrac 1 3 \times \begin {vmatrix} \dfrac 1 2 & \dfrac 1 3 \\ \dfrac 1 3 & \dfrac 1 4 \end {vmatrix}\)
Expansion Theorem for Determinants
\(\ds \) \(=\) \(\ds 1 \times \paren {\dfrac 1 3 \times \dfrac 1 5 - \dfrac 1 4 \times \dfrac 1 4} 
- \dfrac 1 2 \times \paren {\dfrac 1 2 \times \dfrac 1 5 - \dfrac 1 4 \times \dfrac 1 3}
+ \dfrac 1 3 \times \paren {\dfrac 1 2 \times \dfrac 1 4 - \dfrac 1 3 \times \dfrac 1 3}\)
Expansion Theorem for Determinants
\(\ds \) \(=\) \(\ds \paren {\dfrac 1 {15} - \dfrac 1 {16} } 
- \dfrac 1 2 \times \paren {\dfrac 1 {10} - \dfrac 1 {12} }
+ \dfrac 1 3 \times \paren {\dfrac 1 8 - \dfrac 1 9}\)
Multiplication of Fractions
\(\ds \) \(=\) \(\ds \dfrac {16 - 15} {15 \times 16} - \dfrac {12 - 10} {2 \times 10 \times 12} + \dfrac {9 - 8} {3 \times 8 \times 9}\) Addition of Fractions and Multiplication of Fractions
\(\ds \) \(=\) \(\ds \dfrac 1 {240} - \dfrac 1 {120} + \dfrac 1 {216}\) simplification
\(\ds \) \(=\) \(\ds \dfrac 1 {216} - \dfrac 1 {240}\) simplification, noting that $\dfrac 1 {240} = \dfrac 1 2 \times \dfrac 1 {120}$
\(\ds \) \(=\) \(\ds \dfrac {240 - 216} {216 \times 240}\) Addition of Fractions
\(\ds \) \(=\) \(\ds \dfrac {24} {51 \, 840}\) simplification
\(\ds \) \(=\) \(\ds \dfrac 1 {2160}\) dividing top and bottom by $24$

$\blacksquare$


Sources

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