Question

Solve the following equations by the method of inversion. x + y = 1, y + z = 5/3, z + x = 4/3. [ Find A-¹ using row transformations. ]​

Answer 1

Given : x+y=1, y+z=5/3, z+x=4/3​

To Find : Solve using method of inversion

Solution:

x + y =    1

y + z  =   5/3

x  + z  =  4/3

Adding all 2(x + y + z)  = 4

=>  x + y + z = 2

=> z = 2 - 1 = 1

   x = 2 - 5/3 = 1/3

   y = 2 - 4/3 = 2/3

$\left[\begin{array}{ccc}1&1&0\\0&1&1\\1&0&1\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}1\\5/3\\4/3\end{array}\right]$

| A | = 1 (1 - 0 ) - 1(0 - 1)  + 0(0 - 1)

= 1 + 1 + 0

= 2

 $Adj A = \left[\begin{array}{ccc}1&-1&1\\1&1&-1\\-1&1&1\end{array}\right]$

A⁻¹  = Adj A / | A |

$\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \dfrac{1}{2} \left[\begin{array}{ccc}1&-1&1\\1&1&-1\\-1&1&1\end{array}\right] \left[\begin{array}{ccc}1\\5/3\\4/3\end{array}\right]$

$\left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \dfrac{1}{2} \left[\begin{array}{ccc}2/3\\4/3\\2\end{array}\right]$

x = 1/3

y = 2/3

z = 1

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