Determinant.
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Determinant.
Determinant, mathematical notation consisting of a square array of numbers or other elements between two vertical bars; the value of the expression is determined by
its expansion according to certain rules. Determinants were first investigated by the Japanese mathematician Takakazu Seki(also known as Seki Kowa) about 1683 and
independently by the German philosopher and mathematician Gottfried Wilhelm Leibniz about 1693. These notations are used in almost every branch of mathematics
and in the natural sciences.
The symbol
is a determinant of the second order, because it is an array of two rows and two columns. Each letter a stands for a number or variable. The
determinant itself also represents a number or variable, the value of which is defined as a11 a22-a12a21. For example:
A determinant of the n th order is a square array of n rows and n columns represented by the symbol
The minor, Mij, of any element aij in the array is the determinant formed of the elements remaining after deleting the row i and the column j in which the element aij
occurs. The cofactor, Aij, of an element aij is equal to (- 1)i+ jMij.
The value of any determinant may be expressed in terms of the elements of any row (or column) and their respective cofactors in accordance with the following rule.
Each element in the selected row (or column) is multiplied by its corresponding cofactor; the sum of these products is the value of the determinant. Formally, this may
be written
if the expansion is in terms of the ith row, or
if it is in terms of the jth column. Thus, to find the value of a third-order determinant using the elements in the first column
These terms may be evaluated in accordance with the definition of the second-order determinant given above. For determinants of higher orders than the third, the
process is repeated on the determinants formed by the minors until the determinants can be expanded easily.
Because this method of finding the value of a determinant may be quite laborious, various properties of a determinant are developed and utilized to lessen the amount
of calculation needed to evaluate it. Among these properties are the following: (1) a determinant is equal to zero if all the elements in one row (or column) are identical
with or proportional to the elements in another row (or column); (2) a determinant is multiplied by a given factor if each element of a row (or column) is multiplied by
the same factor; and (3) the value of a determinant is not changed by adding to each element of a row (or column) the corresponding element of another row (or
column) multiplied by a constant factor. Hence, through the use of these and other properties, determinants of higher order can be reduced to third-order determinants
for simple expansion.
Application of determinants in analytical geometry is illustrated in the following example: If P1(x1, y1), P2(x2, y2), P3(x3, y3) are three distinct points in a rectangular
coordinate plane, the area A of triangle P1P2P3, apart from algebraic sign, is given by
When the three points are collinear, the determinant is equal to zero.
An example of the use of determinants in solving linear equations is as follows. Let
be a system of n linear equations in the n unknowns x1, x, ..., xn. The determinant ? given above is the determinant of coefficients; let ?k be the determinant obtained
by deleting the kth column of ? and replacing it by the column of constants b1, b2, ..., bn, where k = 1, 2, ..., n. If ?? 0, the equations are consistent--that is, a
solution is possible. In this case only one solution is possible; it is given by
If ? = 0, further investigation is necessary to determine the number and nature of the solutions.
Contributed By:
James Singer
Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
Determinant.
Determinant, mathematical notation consisting of a square array of numbers or other elements between two vertical bars; the value of the expression is determined by
its expansion according to certain rules. Determinants were first investigated by the Japanese mathematician Takakazu Seki(also known as Seki Kowa) about 1683 and
independently by the German philosopher and mathematician Gottfried Wilhelm Leibniz about 1693. These notations are used in almost every branch of mathematics
and in the natural sciences.
The symbol
is a determinant of the second order, because it is an array of two rows and two columns. Each letter a stands for a number or variable. The
determinant itself also represents a number or variable, the value of which is defined as a11 a22-a12a21. For example:
A determinant of the n th order is a square array of n rows and n columns represented by the symbol
The minor, Mij, of any element aij in the array is the determinant formed of the elements remaining after deleting the row i and the column j in which the element aij
occurs. The cofactor, Aij, of an element aij is equal to (- 1)i+ jMij.
The value of any determinant may be expressed in terms of the elements of any row (or column) and their respective cofactors in accordance with the following rule.
Each element in the selected row (or column) is multiplied by its corresponding cofactor; the sum of these products is the value of the determinant. Formally, this may
be written
if the expansion is in terms of the ith row, or
if it is in terms of the jth column. Thus, to find the value of a third-order determinant using the elements in the first column
These terms may be evaluated in accordance with the definition of the second-order determinant given above. For determinants of higher orders than the third, the
process is repeated on the determinants formed by the minors until the determinants can be expanded easily.
Because this method of finding the value of a determinant may be quite laborious, various properties of a determinant are developed and utilized to lessen the amount
of calculation needed to evaluate it. Among these properties are the following: (1) a determinant is equal to zero if all the elements in one row (or column) are identical
with or proportional to the elements in another row (or column); (2) a determinant is multiplied by a given factor if each element of a row (or column) is multiplied by
the same factor; and (3) the value of a determinant is not changed by adding to each element of a row (or column) the corresponding element of another row (or
column) multiplied by a constant factor. Hence, through the use of these and other properties, determinants of higher order can be reduced to third-order determinants
for simple expansion.
Application of determinants in analytical geometry is illustrated in the following example: If P1(x1, y1), P2(x2, y2), P3(x3, y3) are three distinct points in a rectangular
coordinate plane, the area A of triangle P1P2P3, apart from algebraic sign, is given by
When the three points are collinear, the determinant is equal to zero.
An example of the use of determinants in solving linear equations is as follows. Let
be a system of n linear equations in the n unknowns x1, x, ..., xn. The determinant ? given above is the determinant of coefficients; let ?k be the determinant obtained
by deleting the kth column of ? and replacing it by the column of constants b1, b2, ..., bn, where k = 1, 2, ..., n. If ?? 0, the equations are consistent--that is, a
solution is possible. In this case only one solution is possible; it is given by
If ? = 0, further investigation is necessary to determine the number and nature of the solutions.
Contributed By:
James Singer
Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
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