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Determinant


Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. As shown by Cramer's rule, a nonhomogeneous system of linear equations has a unique solution iff the determinant of the system's matrix is nonzero (i.e., the matrix is nonsingular). For example, eliminating x, y, and z from the equations

a_1x+a_2y+a_3z=0
(1)
b_1x+b_2y+b_3z=0
(2)
c_1x+c_2y+c_3z=0
(3)

gives the expression

 a_1b_2c_3-a_1b_3c_2+a_2b_3c_1-a_2b_1c_3+a_3b_1c_2-a_3b_2c_1=0,
(4)

which is called the determinant for this system of equation. Determinants are defined only for square matrices.

If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, the matrix is said to be unimodular.

The determinant of a matrix A,

 |a_1 a_2 ... a_n; b_1 b_2 ... b_n; | | ... |; z_1 z_2 ... z_n|
(5)

is commonly denoted det(A), |A|, or in component notation as sum(+/-a_1b_2c_3...), D(a_1b_2c_3...), or |a_1b_2c_3...| (Muir 1960, p. 17). Note that the notation det(A) may be more convenient when indicating the absolute value of a determinant, i.e., |det(A)| instead of ||A||. The determinant is implemented in the Wolfram Language as Det[m].

A 2×2 determinant is defined to be

 det[a b; c d]=|a b; c d|=ad-bc.
(6)

A k×k determinant can be expanded "by minors" to obtain

 |a_(11) a_(12) a_(13) ... a_(1k); a_(21) a_(22) a_(23) ... a_(2k); | | | ... |; a_(k1) a_(k2) a_(k3) ... a_(kk)|=a_(11)|a_(22) a_(23) ... a_(2k); | | ... |; a_(k2) a_(k3) ... a_(kk)| 
 -a_(12)|a_(21) a_(23) ... a_(2k); | | ... |; a_(k1) a_(k3) ... a_(kk)|+...+/-a_(1k)|a_(21) a_(22) ... a_(2(k-1)); | | ... |; a_(k1) a_(k2) ... a_(k(k-1))|.
(7)

A general determinant for a matrix A has a value

 |A|=sum_(i=1)^ka_(ij)C_(ij),
(8)

with no implied summation over j and where C_(ij) (also denoted a^(ij)) is the cofactor of a_(ij) defined by

 C_(ij)=(-1)^(i+j)M_(ij).
(9)

and M_(ij) is the minor of matrix A formed by eliminating row i and column j from A. This process is called determinant expansion by minors (or "Laplacian expansion by minors," sometimes further shortened to simply "Laplacian expansion").

A determinant can also be computed by writing down all permutations of {1,...,n}, taking each permutation as the subscripts of the letters a, b, ..., and summing with signs determined by epsilon_p=(-1)^(i(p)), where i(p) is the number of permutation inversions in permutation p (Muir 1960, p. 16), and epsilon_(n_1n_2...) is the permutation symbol. For example, with n=3, the permutations and the number of inversions they contain are 123 (0), 132 (1), 213 (1), 231 (2), 312 (2), and 321 (3), so the determinant is given by

 |a_1 a_2 a_3; b_1 b_2 b_3; c_1 c_2 c_3| =a_1b_2c_3-a_1b_3c_2-a_2b_1c_3+a_2b_3c_1+a_3b_1c_2-a_3b_2c_1.
(10)

If a is a constant and A an n×n square matrix, then

 |aA|=a^n|A|.
(11)

Given an n×n determinant, the additive inverse is

 |-A|=(-1)^n|A|.
(12)

Determinants are also distributive, so

 |AB|=|A||B|.
(13)

This means that the determinant of a matrix inverse can be found as follows:

 |I|=|AA^(-1)|=|A||A^(-1)|=1,
(14)

where I is the identity matrix, so

 |A|=1/(|A^(-1)|).
(15)

Determinants are multilinear in rows and columns, since

 |a_1 a_2 a_3; a_4 a_5 a_6; a_7 a_8 a_9|=|a_1 0 0; a_4 a_5 a_6; a_7 a_8 a_9|+|0 a_2 0; a_4 a_5 a_6; a_7 a_8 a_9|+|0 0 a_3; a_4 a_5 a_6; a_7 a_8 a_9|
(16)

and

 |a_1 a_2 a_3; a_4 a_5 a_6; a_7 a_8 a_9|=|a_1 a_2 a_3; 0 a_5 a_6; 0 a_8 a_9|+|0 a_2 a_3; a_4 a_5 a_6; 0 a_8 a_9|+|0 a_2 a_3; 0 a_5 a_6; a_7 a_8 a_9|.
(17)

The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix

|BAB^(-1)|=|B||A||B^(-1)|
(18)
=|B||A|1/(|B|)
(19)
=|A|.
(20)

The determinant of a similarity transformation minus a multiple of the unit matrix is given by

|B^(-1)AB-lambdaI|=|B^(-1)AB-B^(-1)lambdaIB|
(21)
=|B^(-1)(A-lambdaI)B|
(22)
=|B^(-1)||A-lambdaI||B|
(23)
=|A-lambdaI|.
(24)

The determinant of a transpose equals the determinant of the original matrix,

 |A|=|A^(T)|,
(25)

and the determinant of a complex conjugate is equal to the complex conjugate of the determinant

 |A^_|=|A|^_.
(26)

Let epsilon be a small number. Then

 |I+epsilonA|=1+epsilonTr(A)+O(epsilon^2),
(27)

where Tr(A) is the matrix trace of A. The determinant takes on a particularly simple form for a triangular matrix

 |a_(11) a_(21) ... a_(k1); 0 a_(22) ... a_(k2); | | ... |; 0 0 ... a_(kk)|=product_(n=1)^ka_(nn).
(28)

Important properties of the determinant include the following, which include invariance under elementary row and column operations.

1. Switching two rows or columns changes the sign.

2. Scalars can be factored out from rows and columns.

3. Multiples of rows and columns can be added together without changing the determinant's value.

4. Scalar multiplication of a row by a constant c multiplies the determinant by c.

5. A determinant with a row or column of zeros has value 0.

6. Any determinant with two rows or columns equal has value 0.

Property 1 can be established by induction. For a 2×2 matrix, the determinant is

|a_1 b_1; a_2 b_2|=a_1b_2-b_1a_2
(29)
=-(b_1a_2-a_1b_2)
(30)
=-|b_1 a_1; b_2 a_2|
(31)

For a 3×3 matrix, the determinant is

 |a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3|=a_1|b_2 c_2; b_3 c_3|-b_1|a_2 c_2; a_3 c_3|+c_1|a_2 b_2; a_3 b_3| 
=-(a_1|c_2 b_2; c_3 b_3|+b_1|a_2 c_2; a_3 c_3|-c_1|a_2 b_2; a_3 b_3|)=-|a_1 c_1 b_1; a_2 c_2 b_2; a_3 c_3 b_3| 
=-(-a_1|b_2 c_2; b_3 c_3|+b_1|a_2 c_2; a_3 c_3|+c_1|b_2 a_2; b_3 a_3|)=-|b_1 a_1 c_1; b_2 a_2 c_2; b_3 a_3 c_3| 
=-(a_1|c_2 b_2; c_3 b_3|-b_1|c_2 a_2; c_3 a_3|+c_1|b_2 a_2; b_3 a_3|)=-|c_1 b_1 a_1; c_2 b_2 a_2; c_3 b_3 a_3|.
(32)

Property 2 follows likewise. For 2×2 and 3×3 matrices,

|ka_1 b_1; ka_2 b_2|=k(a_1b_2)-k(b_1a_2)
(33)
=k|a_1 b_1; a_2 b_2|
(34)

and

|ka_1 b_1 c_1; ka_2 b_2 c_2; ka_3 b_3 c_3|=ka_1|b_2 c_2; b_3 c_3|-b_1|ka_2 c_2; ka_3 c_3|+c_1|ka_2 b_2; ka_3 b_3|
(35)
=k|a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3|.
(36)

Property 3 follows from the identity

 |a_1+kb_1 b_1 c_1; a_2+kb_2 b_2 c_2; a_3+kb_3 b_3 c_3|=(a_1+kb_1)|b_2 c_2; b_3 c_3|-b_1|a_2+kb_2 c_2; a_3+kb_3 c_3|+c_1|a_2+kb_2 b_2; a_3+kb_3 b_3|.
(37)

If a_(ij) is an n×n matrix with a_(ij) real numbers, then det[a_(ij)] has the interpretation as the oriented n-dimensional content of the parallelepiped spanned by the column vectors [a_(i,1)], ..., [a_(i,n)] in R^n. Here, "oriented" means that, up to a change of + or - sign, the number is the n-dimensional content, but the sign depends on the "orientation" of the column vectors involved. If they agree with the standard orientation, there is a + sign; if not, there is a - sign. The parallelepiped spanned by the n-dimensional vectors v_1 through v_i is the collection of points

 t_1v_1+...+t_iv_i,
(38)

where t_j is a real number in the closed interval [0,1].

Several accounts state that Lewis Carroll (Charles Dodgson) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants. Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland, expressed a desire to receive the author's next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author's other books, and was sent one of Dodgson's mathematical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worth while to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson).

DetComplexMatrix

Hadamard (1893) showed that the absolute value of the determinant of a complex n×n matrix with entries in the unit disk satisfies

 |detA|<=n^(n/2)
(39)

(Brenner 1972). The plots above show the distribution of determinants for random n×n complex matrices with entries satisfying |a_(ij)|<1 for n=2, 3, and 4.


See also

Cayley-Menger Determinant, Chió Pivotal Condensation, Circulant Determinant, Cofactor, Condensation, Cramer's Rule, Determinant Expansion by Minors, Determinant Identities, Elementary Row and Column Operations, Hadamard's Maximum Determinant Problem, Hessian, Hyperdeterminant, Immanant, Jacobian, Knot Determinant, Matrix, Minor, Permanent, Pfaffian, Singular Matrix, Sylvester's Determinant Identity, Sylvester Matrix, System of Equations, Unimodular Matrix, Vandermonde Determinant, Wronskian Explore this topic in the MathWorld classroom

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References

Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158, 1-14, 1993.Arfken, G. "Determinants." §4.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 168-176, 1985.Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Dodgson, C. L. An Elementary Treatise on Determinants, with Their Application to Simultaneous Linear Equations and Algebraical Geometry. London: Macmillan, 1867.Dostor, G. Eléments de la théorie des déterminants, avec application à l'algèbre, la trigonométrie et la géométrie analytique dans le plan et l'espace, 2ème ed. Paris: Gauthier-Villars, 1905.Gattegno, J. Lewis Carroll: Fragments of a Looking-Glass. New York: Crowell, 1974.Hadamard, J. "Résolution d'une question relative aux déterminants." Bull. Sci. Math. 17, 30-31, 1893.Heath, P. The Philosopher's Alice: Alice's Adventures in Wonderland and Through the Looking-Glass. New York: St. Martin's Press, 1974.Kowalewski, G. Einführung in die Determinantentheorie. New York: Chelsea, 1948.Mikkelson, D. P. and Mikkelson, B. "Fit for a Queen." http://www.snopes.com/language/literary/carroll.htm.Muir, T. A Treatise on the Theory of Determinants. New York: Dover, 1960.Whittaker, E. T. and Robinson, G. "Determinants and Linear Equations." Ch. 5 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 71-77, 1967.Yvinec, Y. "Geometric Computing: Exact Sign of a Determinant." http://www-sop.inria.fr/prisme/personnel/yvinec/Determinants/english.html.

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Determinant

Cite this as:

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

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