Cramer's Rule

Given the system of equation;

Then

ex.
solve the system:
7x - 5y = -50
2x + y = -7

Third Order Determinants, are 3×3 determinants, which is written as,

The minor of an element in a determinant is a determinant that is obtained after the row and column in which the element appears are deleted.

ex.
Given the determinant; find the minor of a₁

To calculate third order determinants, a method called expansion of a determinant by minors can be used.

This process is called expansion of a determinant by minors about the first column.

It is possible to expand a determinant about any row or any column, provided that the right sign is applied to the terms of the expansion. The following sign array is very useful in determining the signs of the term of the expansion.

ex.
Expand the determinant by minors about the second column.
The second column in the sign array is - + -.

= -7( 5 - 2) -9(3 - 4) -6(-3 -10)
= -7(3) -9(-1) -6(-13)
= -21 + 9 + 78
= 66

Expanded Cramer's Rule:

Given the system of equation;

Then

ex.
5x -2y + 3z = -1
3x + y - 2z = 25
2x - 4y + 5z = 16

= 5(5-8) + 2(15-(-4)) + 3(-12 -2)
= 5(-3) + 2(19) + 3(-14)
= -15 + 38 - 42
= -19
D = -19

= -1(5-8) -25(-10 - (-12)) - 29(4 - 3)
= -1(-3) -25(2) -29
= 3 - 50 -29
D_x= -76

= 1(15 - (-4)) + 25 (25 -6) + 29 (-10 -9)
= 19 + 25(19) + 29(-19)
= 19( 1 + 25 - 29)
= 19(-3)
D_y = -57

= -1(-12 - 2) -25( -20 - (-4)) - 29(5-(-6))
= -1(-14) -25(-16) -29(11)
= 14 + 400 - 319
= 95
D_z = 95

NOTE: The expansion by minors method can be used with larger order of determinants, but it becomes more tedious as the determinant gets bigger.