Determinant

In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.

General definition and computation

Suppose A = (A_{i,j}) , is a square matrix.

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If A is a 1-by-1 matrix, then det(A) = A_{1,1} ,

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If A is a 2-by-2 matrix, then det(A) = A_{1,1}A_{2,2} - A_{2,1}A_{1,2} ,

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For a 3-by-3 matrix A, the formula is more complicated:

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:

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egin{matrix}

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det(A) & = & A_{1,1}A_{2,2}A_{3,3} + A_{1,3}A_{2,1}A_{3,2} + A_{1,2}A_{2,3}A_{3,1}\

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& & - A_{1,3}A_{2,2}A_{3,1} - A_{1,1}A_{2,3}A_{3,2} - A_{1,2}A_{2,1}A_{3,3}.

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end{matrix},

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For a general n-by-n matrix, the determinant was defined by Gottfried Leibniz with what is now known as the Leibniz formula:

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:det(A) = sum_{sigma in S_n}

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sgn(sigma) prod_{i=1}^n A_{i, sigma(i)}

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The sum is computed over all permutations sigma of the numbers {1,2,...,n} and sgn(sigma) denotes the signature of the permutation sigma: +1 if sigma is an even permutation and −1 if it is odd (see even and odd permutations).

Related Topics:
Permutation - Signature - Even and odd permutations

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This formula contains n! (factorial) summands and is therefore impractical to use it to calculate determinants for large n.

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In general, determinants can be computed with the Gauss algorithm using the following rules:

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• If $A$ is a triangular matrix, i.e. $A\_\{i,j\}\; =\; 0\; ,$ whenever $i\; >\; j$, then $det(A)\; =\; A\_\{1,1\}\; A\_\{2,2\}\; cdots\; A\_\{n,n\}\; ,$
• If $B$ results from $A$ by exchanging two rows or columns, then $det(B)\; =\; -det(A)\; ,$
• If $B$ results from $A$ by multiplying one row or column with the number $c$, then $det(B)\; =\; c,det(A)\; ,$
• If $B$ results from $A$ by adding a multiple of one row to another row, or a multiple of one column to another column, then $det(B)\; =\; det(A)\; ,$

Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.

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It is also possible to expand a determinant along a row or column using Laplace's formula, which is efficient for relatively small matrices. To do this along row i, say, we write

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:det(A) = sum_{j=1}^n A_{i,j}C_{i,j} = sum_{j=1}^n A_{i,j} (-1)^{i+j} M_{i,j}

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where the C_{i,j} represent the matrix cofactors, i.e. C_{i,j} is (-1)^{i+j} times the minor M_{i,j}, which is the determinant of the matrix that results from A by removing the i-th row and the j-th column.

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