The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

Home » The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

Solution:

Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ .
As per the question,
$\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$
Now converting the following equations in matrix form,
$\begin{array}{l} A X=B \\ {\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 5 & 1 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 2 \\ 1 \\ 6 \end{array}\right]} \\ R_{2}-R_{1} \\ R_{3}-R_{1} \\ {\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 4 & 0 & 0 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 2 \\ -1 \\ 5 \end{array}\right]} \end{array}$
Again converting back into the equations we get
$\begin{array}{l} x+y+z=2 \\ Y=-1 \\ 4 x=5 \\ x=\frac{5}{4} \\ \frac{5}{4}-1+z=2 \\ Z=2-\frac{5}{4}+1 \\ Z=\frac{7}{4} \end{array}$
$\therefore$ The numbers are $\frac{5}{4}, \frac{7}{4},-1$ .