Row reduction method

Row reduction is a method used to solve systems of linear equations by transforming the augmented matrix into row echelon form and then into reduced row echelon form. This process involves a series of row operations that do not change the solutions of the system of equations.

A general form of a system of three unknown simultaneous equations in variables x, y, and z  has the form.

\(\begin{aligned}
& a_1 x+b_1 y+c_1 z=d_1 \\
& a_2 x+b_2 y+c_2 z=d_2 \\
& a_3 x+b_3 y+c_3 z=d_3
\end{aligned} \)

 The above equations can be written in augmented matrix form as

\( \left[\begin{array}{lll|l}
a_1 & b_1 & c_1 & d_1 \\
a_2 & b_2 & c_2 & d_2 \\
a_3 & b_3 & c_3 & d_3
\end{array}\right]\)

Using row operations, we must reduce the above-augmented matrix form to the echelon form as

\( \left[\begin{array}{lll|l}
a & b & c & d \\
0 & e & f & g \\
0 & 0 & h & i
\end{array}\right]\)

[a] Unique solution

We arrive at a unique solution If \( h \neq 0\) we can determine \(z\) uniquely using \(z=\frac{i}{h}\) , and likewise \(y\) and \(x\) from the other two rows.

[b] No solution

The system has no solutions if \(h=0 \text { and } i \neq 0 \)

[c] Infinitely many solutions

The system of equations has no solutions if \( h=0 \text { and } i=0\),  the last row terms all become zero. 

The infinitely many solutions are of the form \( x=p+k t, \quad y=q+l t, \quad \text { and } \quad z=t \quad \text { where } t \in \mathbb{R}\)

Question No: 1