## determinant
determinant, a polynomial expression that is inherent in the entries of a square matrix. The size i and j each take on the values 1, 2, 3, … n. Its non-vanishing detects invertibility of the matrix. Its value is the sum of all terms S(π) a 1π(1) … a , where π ranges over all permutations of (1, 2, … n π( n )n ) and S(π) = ±1 is a sign called the signature of π. This value may be found more easily by expanding the determinant by minors. The minor A of an element ij a of an ij n th-order determinant is the determinant of order ( n - 1) formed by deleting the i th row and the j th column of the original determinant. For example, in the determinanta 21, whose value is 3, has the minorIn expanding a determinant by minors, first the minor of every element in a particular row or column is formed. Products are derived by multiplying each minor by its corresponding element. A plus sign is placed in front of each product if the sum of the row number and column number of its element is even, and a minus sign if the sum is odd. Finally, the signed products are added algebraically. For example, expanding the above determinant by its second row yields:;e12;none;1;e12;;;block;;;;no;1;71808n;174345n;;;;;eq12;comptd;;center;stack;;;;;CE5 Determinants of higher order can be evaluated by successive expansions of this type. By choosing rows of columns containing zeros, some terms can be eliminated. There are various rules for transforming a given determinant, which can be used to obtain a row or column most of whose elements are zeros. Determinants have many applications in mathematics and other fields, e.g., in the solution of simultaneous linear equations.
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