Q.The sum of three numbers is 6. Thrice the third number when added to the first number gives 7. On adding three times first number to the sum of second and third number we get 12. Find the three numbers by using Matrices.

Reduce the augmented matrix using elementary row operations.

Equations: $x+y+z=6$, $y+3z=11$, $x-2y+z=0$ (i.e. $x+z=2y$)

Augmented matrix: $\begin{bmatrix}1&1&1&|&6\\0&1&3&|&11\\1&-2&1&|&0\end{bmatrix}$

$R_3\to R_3-R_1$: $\begin{bmatrix}1&1&1&|&6\\0&1&3&|&11\\0&-3&0&|&-6\end{bmatrix}$

$R_3\to R_3+3R_2$: $\begin{bmatrix}1&1&1&|&6\\0&1&3&|&11\\0&0&9&|&27\end{bmatrix}$

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Reduce the augmented matrix to (near) row-echelon form.

$\begin{bmatrix}1 & 3 & 3\\ 1 & 4 & 4\\ 1 & 3 & 4\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}12\\15\\13\end{bmatrix}$

Augmented matrix: $\begin{bmatrix}1 & 3 & 3 & | & 12\\ 1 & 4 & 4 & | & 15\\ 1 & 3 & 4 & | & 13\end{bmatrix}$

$R_2 \to R_2-R_1$: $\begin{bmatrix}0 & 1 & 1 & | & 3\end{bmatrix}$

$R_3 \to R_3-R_1$: $\begin{bmatrix}0 & 0 & 1 & | & 1\end{bmatrix}$

Reduced system: …

Reduce the augmented matrix $[A\,|\,B]$ to upper-triangular (echelon) form using row operations.

System: $x-y+z=4$; $2x+y-3z=0$; $x+y+z=2$

Augmented form $[A|B] = \left[\begin{array}{ccc|c} 1 & -1 & 1 & 4 \\ 2 & 1 & -3 & 0 \\ 1 & 1 & 1 & 2 \end{array}\right]$

$R_2 \to R_2 - 2R_1$: $\left[\begin{array}{ccc|c} 1 & -1 & 1 & 4 \\ 0 & 3 & -5 & -8 \\ 1 & 1 & 1 & 2 \end{array}\right]$

$R_3 \to R_3 - R_1$: $\left[\begin{array}{ccc|c} 1 & -1 & 1 & 4 \\ 0 & 3 & -5 & -8 \\ 0 & 2 & 0 & -2 \end{array}\right]$

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