Begin2.DVI

Cramer’s Rule

The system of two equations in two unknowns

α 1 x+β 1 y=γ 1

α 2 x+β 2 y=γ 2

or

[

α 1 β 1

α 2 β 2

] [

x

y

]

=

[

γ 1

γ 2

]

has a unique solution ifα 1 β 2 −α 2 β 1 is nonzero. The unique solution is given by

x=

∣∣

∣∣γ^1 β^1

γ 2 β 2

∣∣

∣∣

∣∣

∣∣α^1 β^1

α 2 β 2

∣∣

∣∣

, y=

∣∣

∣∣α^1 γ^1

α 2 γ 2

∣∣

∣∣

∣∣

∣∣α^1 β^1

α 2 β 2

∣∣

∣∣

where

∣∣

∣∣α^1 β^1

α 2 β 2

∣∣

∣∣

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−α 2 β 1

+α 1 β 2

=α 1 β 2 −α 2 β 1

is a single number called the determinant of the coefficients.

The system of three equations in three unknowns

α 1 x+β 1 y+γ 1 z=δ 1

α 2 x+β 2 y+γ 2 z=δ 2

α 3 x+β 3 y+γ 3 x=δ 3

has a unique solution if the determinant of the coefficients

∣∣

∣∣

∣∣

α 1 β 1 γ 1

α 2 β 2 γ 2

α 3 β 3 γ 3

∣∣

∣∣

∣∣=α^1 β^2 γ^3 +β^1 γ^2 α^3 +γ^1 α^2 β^3 −α^3 β^2 γ^1 −β^3 γ^2 α^1 −γ^3 α^2 β^2

is nonzero. A mnemonic device to aid in calculating the determinant of the co-

efficients is to append the first two columns of the coefficients to the end of the

array and then draw diagonals through the coefficients. Multiply the elements along

an arrow and place a plus sign on the products associated withthe down arrows

and a minus sign associated with the products of the up arrows. This gives the figure

∣∣

∣∣

∣∣

α 1 β 1 γ 1

α 2 β 2 γ 2

α 3 β 3 γ 3

∣∣

∣∣

∣∣

α 1 β 1

α 2 β 2

α 3 β 3

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=α 1 β 2 γ 3 +β 1 γ 2 α 3 +γ 1 α 2 β 3 −α 3 β 2 γ 1 −β 3 γ 2 α 1 −γ 3 α 2 β 2

The solution of the three equations, three unknown system ofequations is given

by the determinant ratios

x=

∣∣

∣∣

∣∣

δ 1 β 1 γ 1

δ 2 β 2 γ 2

δ 3 β 3 γ 3

∣∣

∣∣

∣∣

∣∣

∣∣

∣∣

α 1 β 1 γ 1

α 2 β 2 γ 2

α 3 β 3 γ 3

∣∣

∣∣

∣∣

, y=

∣∣

∣∣

∣∣

α 1 δ 1 γ 1

α 2 δ 2 γ 2

α 3 δ 3 γ 3

∣∣

∣∣

∣∣

∣∣

∣∣

∣∣

α 1 β 1 γ 1

α 2 β 2 γ 2

α 3 β 3 γ 3

∣∣

∣∣

∣∣

, z=

∣∣

∣∣

∣∣

α 1 β 1 δ 1

α 2 β 2 δ 2

α 3 β 3 δ 3

∣∣

∣∣

∣∣

∣∣

∣∣

∣∣

α 1 β 1 γ 1

α 2 β 2 γ 2

α 3 β 3 γ 3

∣∣

∣∣

∣∣

and is known as Cramer’s rule for solving a system of equations.