Q.If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third number we get 8 and if three times the first number is added to the sum of second and third number we get 4. Find the numbers 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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