Answer to Question #103629 in Linear Algebra for na

Question

The Gaussian elimination method can only be applied for solving a system of 4
linear equations if the number of variables is 4. Is it true or false ? give reasons.

Answer 1

Solution

It is false. In case the number of linear equations is equal to the number of variables, the system has only one solution. In case the number of linear equations is less than the number of variables, the system has infinitely many solutions.

Consider an arbitrary system of linear equations with n variables and r equations. Take into account the fact that any system of linear equations is equivalent to some stepwise system of linear equations. We can assume that its expanded matrix has the form as:

$\begin{bmatrix} 0 & ... & 0 & a_{1j_1} & a_{1j_2} & ... & a_{1j_r} & ... & a_{1n} & b_1 \\ 0 & ... & 0 & 0 & a_{2j_1} & ... & a_{2j_r} & ... & a_{2n} & b_2 \\ ... & ... &... &... &... & ... &... & ... &... &... \\ 0 & ... &... & 0 &... & ... & a_{rj_r} & ... & a_{rn} & b_r \end{bmatrix}$

Where a_1j1, a_2j2 … a_rjn are not null.

In case r = n expanded matrix has the form as:

$\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n} & b_1 \\ ... & ... &... & ... &... \\ 0 & ... &... & a_{nn} & b_n \end{bmatrix}$

From the last linear equation we can get the value of x_n. Then rising up through the system and substituting the known values of x_i, we can find the values of all variables. This solution will be unique.

In case r > n expanded matrix has the form as:

$\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n} & b_1 \\ ... & ... &... & ... &... \\ 0 & ... &... & a_{rn} & b_r \end{bmatrix}$

Throw free unknowns (x_r+1.. x_n) to the right side. Give them free values from R. Now we can solve the system with respect to the remaining unknowns as it was done in the previous case (r = n). For given values of free unknowns, the remaining unknowns are uniquely determined. In such case the system has infinitely many solutions.

Answer

The Gaussian elimination method can be applied not only to the system where the number of linear equations is equal to the number of variables, but also for the system where the number of linear equations is less than the number of variables. In case the number of linear equations is greater than the number of variables, the solution is reduced to the case with equal amount of variables and equations.