MThambeliyagodage_SLE_C22

\[ \begin{bmatrix} x1 & -2x2 & +x3 & -x4 & =3\\2x1 & -4x2 & +x3 & +x4 & = 2\\x1 & -2x2 & -2x3 & +3x4 & = 1\end{bmatrix} \] ************************************************************************************************************************************************************************************

\[ \begin{bmatrix} 1 & -2 & +1 & -1 & |& 3\\2 & -4 & +1 & +1 & |& 2\\1 & -2 & -2 & +3 & | & 1\end{bmatrix} \text{(add -2 times the 1st row to the 2nd row} \text{.)} \begin{bmatrix} 1 & -2 & +1 & -1 & |& 3\\0 & 0 & -1 & 3 & |& -4\\1 & -2 & -2 & +3 & | & 1\end{bmatrix} \] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & +1 & -1 & |& 3\\0 & 0 & -1 & 3 & |& -4\\1 & -2 & -2 & +3 & | & 1\end{bmatrix} \text{(add -1 times the 1st row to the 3rd row} \text{.)} \begin{bmatrix} 1 & -2 & +1 & -1 & |& 3\\0 & 0 & -1 & 3 & |& -4\\0 & 0 & -3 & +4 & | & -2\end{bmatrix} \] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & -1 & 3 & |& -4\\0 & 0 & -3 & 4 & | & -2\end{bmatrix} \text{(multiply the 2nd row by -1} \text{.)} \begin{bmatrix} 1 & -2 & +1 & -1 & |& 3\\0 & 0 & 1 & -3 & |& 4\\0 & 0 & -3 & 4 & | & -2\end{bmatrix} \] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & -3 & |& 4\\0 & 0 & -3 & 4 & | & -2\end{bmatrix} \text{(add 3 times the 2nd row to the 3rd row} \text{.)} \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & -3 & |& 4\\0 & 0 & 0 & -5 & | & 10\end{bmatrix}\] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & -3 & |& 4\\0 & 0 & 0 & -5 & | & 10\end{bmatrix} \text{(multiply the 3rd row by -1/5} \text{.)} \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & -3 & |& 4\\0 & 0 & 0 & 1 & | & -2\end{bmatrix}\] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & -3 & |& 4\\0 & 0 & 0 & 1 & | & -2\end{bmatrix} \text{(add 3 times the 3rd row to the 2nd row} \text{.)} \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & 0 & |& -2\\0 & 0 & 0 & 1 & | & -2\end{bmatrix}\] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & 1 & -1 & |& 3\\0 & 0 & 1 & 0 & |& -2\\0 & 0 & 0 & 1 & | & -2\end{bmatrix} \text{(add 1 times the 3rd row to the 1st row} \text{.)} \begin{bmatrix} 1 & -2 & 1 & 0 & |& 1\\0 & 0 & 1 & 0 & |& -2\\0 & 0 & 0 & 1 & | & -2\end{bmatrix}\] ************************************************************************************************************************************************************************************ \[ \begin{bmatrix} 1 & -2 & 1 & 0 & |& 1\\0 & 0 & 1 & 0 & |& -2\\0 & 0 & 0 & 1 & | & -2\end{bmatrix} \text{(add -1 times the 2nd row to the 1st row} \text{.)} \begin{bmatrix} 1 & -2 & 0 & 0 & |& 3\\0 & 0 & 1 & 0 & |& -2\\0 & 0 & 0 & 1 & | & -2\end{bmatrix}\] ************************************************************************************************************************************************************************************ \[ \text{Final Reduced Row Matrix =>} \text{} \begin{bmatrix} 1x1 & -2x2 & 0 & 0 & |& 3\\0 & 0 & 1x3 & 0 & |& -2\\0 & 0 & 0 & 1x4 & | & -2\end{bmatrix}\] ************************************************************************************************************************************************************************************ \[ \begin{align}\text{The system has infinitely many solutions : = > } \text{}\begin{matrix} x1&=2x2 + 3 \\x2&=arbitrary\\x3&=-2\\x4&=-2\end{matrix}\end{align}\]