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Determinant Identities

A useful determinant identity allows the following determinant to be expressed using vector operations,

|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|=(x_3-x_1)·[(x_2-x_1)x(x_4-x_3)].
(1)

Additional interesting determinant identities include

|1 a b+c; 1 b c+a; 1 c a+b|=0
(2)

(Muir 1960, p. 39),

|a+b+c+d b c d; b+c+d+a c d a; c+d+a+b d a b; d+a+b+c a b c|=|1 b c d; 1 c d a; 1 d a b; 1 a b c|(a+b+c+d)
(3)

(Muir 1960, p. 41),

|1 a a^2 a^3; 1 b b^2 b^3; 1 c c^2 c^3; 1 d d^2 d^3|=(b-a)(c-a)(c-b)(d-a)(d-b)(d-c)
(4)

(Muir 1960, p. 42),

|bcd a a^2 a^3; cda b b^2 b^3; dab c c^2 c^3; abc d d^2 d^3|=|1 a^2 a^3 a^4; 1 b^2 b^3 b^4; 1 c^2 c^3 c^4; 1 d^2 d^3 d^4|
(5)

(Muir 1960, p. 47),

|0 a^2 b^2 c^2; a^2 0 gamma^2 beta^2; b^2 gamma^2 0 alpha^2; c^2 beta^2 alpha^2 0|=|0 aalpha bbeta cgamma; aalpha 0 cgamma aalpha; bbeta cgamma 0 aalpha; cgamma bbeta aalpha 0|
(6)

(Muir 1960, p. 42),

|1 1 1 1; 1 1+x 1 1; 1 1 1+y 1; 1 1 1 1+z|=xyz
(7)

(Muir 1960, p. 44), and the Cayley-Menger determinant

|0 a b c; a 0 c b; b c 0 a; c b a 0|=|0 1 1 1; 1 0 c^2 b^2; 1 c^2 0 a^2; 1 b^2 a^2 0|
(8)

(Muir 1960, p. 46), which is closely related to Heron's formula.

See also

Determinant

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References

Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960.

Referenced on Wolfram|Alpha

Determinant Identities

Cite this as:

Weisstein, Eric W. "Determinant Identities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeterminantIdentities.html

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