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Cramer’s Rule
Cramer‟s rule is a method of solving a system of linear equations through the use of determinants.
Matrices and Determinants
To use Cramer‟s Rule, some elementary knowledge of matrix algebra is required. An array of numbers, such as
6 5 a 11 a 12 A = 3 4 a 21 a 22
is called a matrix. This is a “2 by 2” matrix. However, a matrix can be of any size, defined by m rows and n columns (thus an “m by n” matrix). A “square matrix,” has the same number of rows as columns. To use Cramer‟s rule, the matrix must be square.
A determinant is number, calculated in the following way for a “2 by 2” matrix:
a 11 a 12 A = = a 11 a 22 - a 21 a 12 a 21 a 22
For example, letting a 11 = 6, a 12 = 5, a 21 = 3, a 22 = 4:
6 5 A= = 6 (4) - 3 (5) = 9 3 4
For “m by n” matrices of orders larger than 2 by 2, there is a general procedure that can be used to find the determinant. This procedure is best explained as an example. Consider the determinant for a 3 by 3 matrix
a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 32 a 33
The determinant A is calculated as follows:
a 22 a 23 a 31 a 23 a 21 a 22 A = a 11 - a 12 + a 13 a 32 a 33 a 31 a 33 a 31 a 32
note the sign change
A = a 11 (a 22 a 33 - a 23 a 32 ) - a 12 (a 21 a 33 - a 23 a 31 ) + a 13 (a 21 a 32 - a 22 a 31 )
Sign change (like a “2 by 2” matrix)
Note : Sign changes alternate, following the order: positive, negative, positive, negative, etc.
The determinant of the 3 by 3 matrix is the sum of three products. The first step is to understand the placement of the elements from the matrix into the determinant equation. This is done by:
- The three products to be summed correspond to the three elements along the top row of the matrix (this would be a 11 , a 12 , a 13 ).
- Now, imagine a line that goes though the top row of elements (see the model below).
- Beginning at a 11 , imagine, too, a line through the first column (Figure 1).
- The 4 remaining elements are used to construct a new “2 by 2” matrix, and the element a 11 is used to form the first of the three parts of the calculation: a 22 a 23 a 11 a 32 a 33
- The same process (follow steps 1-4 above) is then repeated for a 12 and a 13 as seen in figures 2 and 3 respectively, i.e., the top row contains the element used to multiply the new “2 by 2” matrix, and the column which contains the element from the top row is omitted.
a 11 a 12 a 13 a 11 a 12 a 13 a 11 a 21 a 31 a 21 a 22 a 23 a 21 a 22 a 23 a 21 a 22 a 23 a 31 a 32 a 33 a 31 a 32 a 33 a 31 a 32 a 33
Figure 1 Figure 2 Figure 3
variables. Position x has one column and corresponds to the number of endogenous variables in the system. Finally, position d contains the exogenous terms of each linear equation.
Note : The determinant for a matrix must not equal 0 (A 0). If A = 0 then there is no solution, or there are infinite solutions (from dividing by zero). Therefore, A 0. When A 0, then a unique solution exists.
Applying Cramer’s Rule in a 2x2 example
Using Cramer‟s rule to solve for the unknowns in the following linear equations:
2x 1 + 6x 2 = 22
-x 1 + 5x 2 = 53
Then, A x = d
2 6 x 1 22
-1 5 x 2 53
The primary determinant A = = 2 (5) - (-1) 6 = 16 -1 5
We need to construct xi = Ai, for i=1 and for i=2. A
The first special determinant A 1 is found by replacing the first column of the primary matrix with the constant „d‟ column. The new special matrix A 1 now appears as:
22 6 A 1 = 53 5
and solved as a regular matrix determinant,
A 1 = 22 (5) - 53 (6) = -
Likewise, the same procedure is done to find the second special determinant A2,
A 2 =
A 2 = 2 (53) - (-1) (22) = 128
We have now determined:
A = 16
A 1 = -
A 2 = 128
Using:
xi = Ai A
we get, A 1 - x 1 = A = 16 = -13 (Solution)
A 2 128
x 2 = A = 16 = 8 (Solution)
Applying Cramer’s Rule in a 3x3 example
Using Cramer‟s Rule to solve for the unknowns in three linear equations:
5x 1 - 2x 2 + 3x 3 = 16 2x 1 + 3x 2 - 5x 3 = 2 4x 1 - 5x 2 + 6x 3 = 7 Then,
5 -2 3 x 1 16 2 3 -5 x 2 = 2 4 -5 6 x 3 7
