Question

Using matrix method, solve the system of equations 3x + 2y - 2z = 3, x + 2y + 3z = 6 and 2x - y + z = 2.

Answer 1

Given system of equations is
$3x+2y-2z=3x+2y+3z=6$
and $2x-y+z=2$
In the form of AX = B,
$\left[\begin{array}{ccc}3& 2& -2\\ 1& 2& 3\\ 2& -1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ 6\\ 2\end{array}\right]$
For $A^{-1},|A|=|3\left(5\right)-2(1-6)+(-2)(-5)|$
$=|15+10+10|=|35|\ne 0\therefore A_{11}=5,A_{12}=5,A_{13}=-5,A_{21}=0,A_{22}=7,A_{23}=7,A_{31}=10,A_{32}=-11and{A}_{33}=4\therefore adjA={\left|\begin{array}{ccc}5& 5& -5\\ 0& 7& 7\\ 10& -11& 4\end{array}\right|}^T=\left|\begin{array}{ccc}5& 0& 10\\ 5& 7& -11\\ -5& 7& 4\end{array}\right|$
Now, $A^{-1}=\frac{adjA}{|A|}=\frac{1}{35}\left|\begin{array}{ccc}5& 0& 10\\ 5& 7& -11\\ -5& 7& 4\end{array}\right|$
For $X=A^{-1}B$,
$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{35}\left[\begin{array}{ccc}5& 0& 10\\ 5& 7& -11\\ -5& 7& 4\end{array}\right]\left[\begin{array}{c}3\\ 6\\ 2\end{array}\right]=\frac{1}{35}\left[\begin{array}{c}15+20\\ 15+42-22\\ -15+42+8\end{array}\right]=\frac{1}{35}\left[\begin{array}{c}35\\ 35\\ 35\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$
$\therefore$ x = 1, y = 1 and z = 1