Express the following equations in matrix form and solve them by method of reduction.\(x-y+z=1,\ 2x-y=1,\ 3x+3y-4z=2\)Solution : The given equations can be written in matrix form as \(\begin{bmatrix}1 & -1 & 1\\ 2 & -1 & 0\\ 3 & 3 & -4\end{bmatrix}\b…

Q.3 (C) (i) Express the following equations in matrix form and solve them by method of reduction.

\(x-y+z=1,\ 2x-y=1,\ 3x+3y-4z=2\)

Solution : The given equations can be written in matrix form as \(\begin{bmatrix}1 & -1 & 1\\ 2 & -1 & 0\\ 3 & 3 & -4\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ 1\\ 2\end{bmatrix}\)

By \(R_2\to R_2-2R_1\): \(\begin{bmatrix}1 & -1 & 1\\ \square & \square & \square\\ 3 & 3 & -4\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ -1\\ 2\end{bmatrix}\)

By \(R_3\to R_3-3R_1\): \(\begin{bmatrix}1 & -1 & 1\\ 0 & 1 & -2\\ \square & \square & \square\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ -1\\ -1\end{bmatrix}\)

By \(R_3\to R_3-6R_2\): \(\begin{bmatrix}1 & -1 & 1\\ 0 & 1 & -2\\ 0 & 0 & 5\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}1\\ -1\\ \square\end{bmatrix}\)

We write equations as \(x-y+z=1\) .....(I), \(y-2z=-1\) .....(II), \(5z=5\) .....(III). Solving equations (I), (II) and (III) we get \(x=\square,\ y=\square,\ z=\square\)