§3. Systems of linear equations

Solving systems of linear equations on Cramer's rule

The solution to the system

(3)

is given by , i=1,2…,n, where

Provided that Δ≠0.

Notes:

Cramer's rule works on systems that have exactly one solution.

Cramer's rule gives us a precise formula for finding the solution to an independent system.

Note that Δ is the determinant made up of the original coefficients of , i=1,2…,n. Δ is used in the denominator for , i=1,2…,n. is obtained by replacing the first (or ) column of Δ by the constants . is found by replacing the second (or ) column of Δ by the constants and so on is found by replacing the second (or ) column of Δ by the constants .

Solving systems of linear equations on matrix method

Consider a system (3) of n equations with n unknowns. Let us find a solution of system (3), by using matrices.

The matrix method applies only where the number of equations equals that of unknowns. Let us write system (3) in matrix form; for this purpose we introduce, principal matrix А, the column matrix Х, and the column matrix of free terms В:

Then system (3) can be written in the form of the matrix equation АХ=В.

Two matrices of the same size are equal if and only if each element of one matrix equals the corresponding element of the other matrix. To find the matrix Х, we multiply both sides of the matrix equation by the inverse matrix А^-1 on the left .

Since is the identity matrix, we have

.

Thus, to solve the given system of equations by the matrix method, it is sufficient to find the inverse matrix А^-1 and multiply it by В on the right.

Solving systems of linear equations on Gauss's method

Solving a system of n equations with n unknowns by Cramer's rule, we must compute n+1 determinants of order n. This is a hard work.

Moreover, the method of Cramer cannot be used in cases where the principal determinant equals zero or the number of equations does not agree with that of unknowns. In such cases, Gauss' method of successive elimination of unknowns extended by applying matrices is used.

Consider the Gauss method in the case where the number of equations coincides with that of unknowns (3).

Suppose that а₁₁0; let us divide the first equation by this coefficient:

. (*)

Multiplying the resulting equation by –а₂₁ and adding it to the second equation of system (3), we obtain

.

Similarly, multiplying equation (*) by –а_n1 and adding it to the last equation of system (3), we obtain

.

At the end, we obtain the new system of equations with n–1 unknowns:

(4)

System (4) is obtained from system (3) by applying linear transformations of equations; hence this system is equivalent to (3), i.e., any solution of system (4) is a solution of the initial system of equations.

To get rid of х₂ in the third, the forth, …, nth-equation, we multiply the second equation of system (4) by and, multiplying this equation by the negative coefficients of х₂ and summing them, obtain

Performing this procedure n times, we reduce the system of equations to the diagonal form

We determine х_n from the last equation, substitute it in the preceding equation and obtain x_n-1, and so on; going up, we determine х₁ from the first equation. This is the classical Gauss method.