Question

The sum of the three numbers is 6, Twice the third number when added to the first number gives 7. On adding the sum of the second and third to thirce the first number, we get 12. find the numbers, using adjoin method.

Answer 1

Given let a, b, c be the three numbers

Then according to questions,

$a+b+c=6$ ...……….$\left(1\right)$

$a+2c=7$ …………..$\left(2\right)$

$3a+b+c=12$ …………..$\left(3\right)$

Using the determinant property for finding the value of $a,b,c$ we get

$AX=B$

$\Rightarrow X=A^{-1}B$

$\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\\ 3& 1& 1\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}6\\ 7\\ 12\end{array}\right]$ where $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\\ 3& 1& 1\end{array}\right]$

Now $A^{-1}=\frac{adjA}{|A|}$ we know $A=1(0-2)-1(1-6)+1\left(1\right)$ $=-2+5+1$

$|A|=4$

and $adjA$ $=\left[\begin{array}{ccc}-2& 5& 1\\ 0& -2& +2\\ 2& -1& -1\end{array}\right]=\left[\begin{array}{ccc}-2& 0& 2\\ 5& -2& -1\\ 1& 2& -1\end{array}\right]$

Now $\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\frac{1}{4}\left[\begin{array}{ccc}-2& 0& 2\\ 5& -2& -1\\ 1& 2& -1\end{array}\right]\left[\begin{array}{c}6\\ 7\\ 12\end{array}\right]$

$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}-12+0+24\\ 30-14-12\\ 6+14-12\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}12\\ 4\\ 8\end{array}\right]$

Hence $a=3,b=1,c=2$.