Question

Solve by matrix method 2x – y + z = 4, x + y + z = 1, x – 3y – 2z = 2

Answer 1

Terefore x = 1, y = -1 and z = 1

Step-by-step explanation:

Given equations are 2x - y +z=4, x+y + z and x - 3y -2z = 2

$A=\left[\begin{array}{ccc}2&-1&1\\1&1&1\\1&-3&-2\end{array}\right]$          $X = \left[\begin{array}{c}x\\y\\z\end{array}\right]$      and      $B = \left[\begin{array}{c}4\\1\\2\end{array}\right]$

we know that  X = $A^{-1}$× B

$adj A = \left[\begin{array}{ccc}1&-5&-2\\3&-5&-1\\-4&5&3\end{array}\right] \\\\ |A| = 2(-2+3)+1(-2-1)+1(-3-1) = -5$  

$A^{-1}=\frac{adj A}{|A|} = \frac{\left[\begin{array}{ccc}1&-5&-2\\3&-5&-1\\-4&5&3\end{array}\right] }{-5}$

X = $\left[\begin{array}{ccc}\frac{-1}{5} &1&\frac{2}{5}\\ \\\frac{-3}{5} &1&\frac{1}{5}\\ \\\frac{4}{5} &-1&\frac{-3}{5} \end{array}\right]$×$\left[\begin{array}{c}4\\1\\2\end{array}\right]$=$\left[\begin{array}{c}1\\-1\\1\end{array}\right]$

⇔$\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{c}1\\-1\\1\end{array}\right]$

⇔x = 1, y = -1 and z = 1