What is a linear equation?
Before introducing Kramer's law, we need to know about linear equations. A system of linear equations is a set of multiple linear equations, each of which has the following form:
a 1x 1+a2x2+...+anxn=b
Where a 1, a2, ..., an is constant, x 1, x2, ..., xn is unknown, and b is constant. In a linear system of equations, there are many such equations, and the unknowns of each equation may be different, but the forms of the equations are all the same.
For example, the following is a system of linear equations with two equations:
2x+3y=8
4x-5y=-7
In this example, the unknowns are X and Y, the constants are 8 and -7, and the coefficients are 2, 3, 4 and -5.
What is Cramer's Law?
Cramer's rule is a method to solve linear equations by algebraic method. Its basic idea is that for an n-element linear equation system, if the coefficients and constants of each unknown are put in an nxn matrix, then the equation system can be solved by calculating the determinant of this matrix.
For example, for the following linear equation:
2x+3y=8
4x-5y=-7
You can put coefficients and constants into a matrix:
|2? 3|_|x|_|8|
|4-5|x|y|=|-7|
Then calculate the determinant of this matrix:
|2? 3|
|4-5|=(2x-5)-(3x4)=-23
Next, replace the coefficients and constants of each unknown with b to get the following three matrices:
|b？ 3|_|8|_|-23|
|b-5|8|y|=|-23|
Then the determinants of these three matrices are calculated respectively, and the following results are obtained:
|b？ 3|
|b-5|=- 13b-24
|2_|
|4b|=2b- 12
|2? 3|
|4b|=-23- 12b
Finally, divide the value of each determinant by the determinant of the original matrix to get the value of each unknown:
x=(2b- 12)/-23
y=(- 13b-24)/-23
This is the basic step of solving linear equations with Cramer's law.
Advantages and disadvantages of Kramer's law
The advantage of Kramer's law is that it is very intuitive and easy to understand. It does not need other methods such as matrix or Gaussian elimination, so it can solve linear equations faster in some cases.
However, Kramer's law also has some shortcomings. First of all, it is only suitable for small linear equations, because the time complexity of calculating determinant is O(n! )。 Secondly, the calculation of determinant requires a lot of repeated calculations, which is inefficient in large linear equations. Finally, if the determinant of the original matrix is 0, Kramer's rule will not be able to solve the linear equations.