Cramer’s Rule

Topics: Howard Staunton, System of linear equations, Determinant Pages: 11 (1982 words) Published: December 9, 2013
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 a11 a­12
A = 
3 4 a21 a22

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:

a11 a12
A = = a11 a22 - a21 a12
a21 a22

For example, letting a­11­ = 6, a­12 = 5, a­21­ = 3, a­22­ = 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

a11 a12 a13
A = a21 a22 a23
a31 a32 a33

The determinant A is calculated as follows:

a22 a23 a31 a23 a21 a22
A = a11 - a12 + a13
a32 a33 a31 a33 a31 a32

note the sign change

A = a11 (a22 a33 - a23 a32) - a12 (a21 a33 - a23 a31) + a13 (a21 a32 - a22 a31)

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:

1. The three products to be summed correspond to the three elements along the top row of the matrix (this would be a11, a12, a13).

2. Now, imagine a line that goes though the top row of elements (see the model
below).

3. Beginning at a11, imagine, too, a line through the first column (Figure 1).

4. The 4 remaining elements are used to construct a new “2 by 2” matrix, and the element a11 is used to form the first of the three parts of the calculation:
a22 a23
a11
a32 a33

5. The same process (follow steps 1-4 above) is then repeated for a12 and a13 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.

a11 a12 a13a11 a12 a13a11 a21 a31
a21 a22 a23a21 a22 a23a21 a22 a23
a31 a32 a33a31 a32 a33a31 a32 a33

Figure 1 Figure 2 Figure 3

For an example, consider:

5 6 7
A = 2 1 4
9 6 3

Find the determinant A.

Determinant A is calculated as follows:

1 4 2 4 2 1
A = 5 - 6 + 7
6 3 9 3 9 6

 A = 5 [1 (3) - 6 (4)] - 6 [2 (3) - 9 (4) ] + 7 [2 (6) - 9 (1)]

A = 96

A Description of Cramer’s Rule

Cramer’s rule is a method of solving a system of linear equations through the use of determinants. Cramer’s rule is given by the equation

xi = Ai
A

where xi is the i th endogenous variable in a system of equations, A is the determinant of the original A matrix as discussed in the previous section, and Ai is the determinant a special matrix formed as part of Cramer’s rule.

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