Question

The sum of three numbers is 4. If twice the second number is added to the sim of first and third, the sum is 8. adding second & third number to two times the first number, we get 1. find the three numbers using matric method. Use elementary row transformations.

Answer 1

Given

Let a, b, c are three numbers then

$a+b+c=4$ ……$\left(1\right)$

$a+2b+c=8$ ………..$\left(2\right)$

$2a+b+c=1$ ………..$\left(3\right)$

using matrix method $Ax=B$

$\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 2& 1& 1\end{array}\right]=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}4\\ 8\\ 1\end{array}\right]$

Where

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 2& 1& 1\end{array}\right]$

$\Rightarrow |A|=1(2-1)-1(1-2)+1(1-4)$

$|A|=1+1-3=-1$

$[|A|=-1]$

Now adjoint of A is $adjA=\left[\begin{array}{ccc}1& 0& -1\\ -1& -1& 0\\ -3& +1& 1\end{array}\right]$

Now for finding values of a, b, c we have $x=A^{-1}B$

$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$ $=\frac{adjA}{|A|}$$=-1\left[\begin{array}{ccc}1& 0& -1\\ 1& -1& 0\\ -3& 1& 1\end{array}\right]\left[\begin{array}{c}4\\ 8\\ 1\end{array}\right]$

$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=-1\left[\begin{array}{c}4-1\\ 4-8\\ -12+8+1\end{array}\right]=-1\left[\begin{array}{c}3\\ -4\\ -3\end{array}\right]=\left[\begin{array}{c}-3\\ 4\\ 3\end{array}\right]$

Hence $[a=-3,b=4,$ & $c=3]$.