Cramer's method and its application

Cramer's method is one of the exact methods for solving systems of linear algebraic equations (SLAE). Its accuracy is due to the use of determinants of the matrix of the system, as well as certain restrictions imposed in the course of the proof of the theorem.

A system of linear algebraic equations with coefficients belonging, for example, to the set of R-real numbers, from the unknowns x1, x2, ..., xn is a set of expressions of the form

Ai2 x1 + ai2 x2 + ... ain xn = bi for i = 1, 2, ..., m, (1)

Where aij, bi are real numbers. Each of these expressions is called a linear equation, aij - coefficients for unknowns, bi - free coefficients of equations.

A solution of the system (1) is the n-dimensional vector x ° = (x1 °, x2 °, ..., xn °), which when substituted into the system, instead of the unknowns x1, x2, ..., xn, each of the rows in the system becomes a true equality .

A system is said to be joint if it has at least one solution and is incompatible if its solution set coincides with the empty set.

It must be remembered that in order to find a solution to systems of linear algebraic equations using Cramer's method, the system matrices must be square, which essentially means the same number of unknowns and equations in the system.

So, in order to use Cramer's method, one must at least know what the matrix of systems of linear algebraic equations is and how it is written out. And secondly, to understand what is called the determinant of the matrix and know the skills of its calculation.

Suppose that you own this knowledge. Great! Then you just have to remember the formulas that determine the method of Cramer. To simplify the memorization, we use the following notation:

• Det is the main determinant of the system matrix;

• Deti is the determinant of the matrix obtained from the matrix of the system if the i-th column of the matrix is replaced by a column vector whose elements are the right-hand sides of systems of linear algebraic equations;

• N is the number of unknowns and equations in the system.

Then the Cramer rule for computing the i-th component xi (i = 1, ... n) of the n-dimensional vector x can be written in the form

Xi = deti / Det, (2).

Det is strictly nonzero.

The uniqueness of the solution of the system when it is compatible ensures that the principal determinant of the system is zero. Otherwise, if the sum (xi), squared, is strictly positive, then the SLAE with the square matrix will be inconsistent. This can happen, in particular, when at least one of the deti is different from zero.

Example 1 . Solve the three-dimensional system of the LAU using Cramer's formulas.
X1 + 2 x2 + 4 x3 = 31,
5 x1 + x2 + 2 x3 = 29,
3 x1 - x2 + x3 = 10.

Decision. We write out the matrix of the system line by line, where Ai is the ith row of the matrix.
A1 = (1 2 4), A2 = (5 1 2), A3 = (3 -1 1 1).
The column of free coefficients b = (31 29 10).

The main determinant of the Det system is
Det = a11 a22 a33 + a12 a23 a31 + a31 a21 a32 - a13 a22 a31 - a11 a32 a23 - a33 a21 a12 = 1 - 20 + 12 - 12 + 2 - 10 = -27.

To calculate det1, we use the substitution a11 = b1, a21 = b2, a31 = b3. Then
Det1 = b1 a22 a33 + a12 a23 b3 + a31 b2 a32 - a13 a22 b3 - b1 a32 a23 - a33 b2 a12 = ... = -81.

Similarly, to calculate det2, we use the substitution a12 = b1, a22 = b2, a32 = b3 and, accordingly, to calculate det3 - a13 = b1, a23 = b2, a33 = b3.
Then you can check that det2 = -108, and det3 = -135.
According to Cramer's formulas, we find x1 = -81 / (-27) = 3, x2 = -108 / (-27) = 4, x3 = -135 / (-27) = 5.

The answer is: x ° = (3,4,5).

Relying on the conditions for the applicability of this rule, Cramer's method of solving linear equation systems can be used indirectly, for example, to investigate the system for a possible number of solutions depending on the magnitude of some parameter k.

Example 2. Determine for which values of the parameter k the inequality | kx - y - 4 | + | x + ky + 4 | <= 0 has exactly one solution.

Decision.
This inequality, by virtue of the definition of the modulus of a function, can be satisfied only if both expressions are simultaneously zero. Therefore, this problem reduces to finding a solution of a linear system of algebraic equations

Kx - y = 4,
X + ky = -4.

The solution of this system is unique if its principal determinant
Det = k ^ {2} + 1 is non-zero. Obviously, this condition is satisfied for all real values of the parameter k.

Answer: for all real values of the parameter k.

To problems of this kind, many practical problems from the field of mathematics, physics or chemistry can also be reduced .