Cramer’s Rule
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
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