Q.An amount of Rs. 5000 is invested in three types of investments, at interest rates 6.7, 7.7, 8% per annum respectively. The total annual income from these investments is Rs. 350/-. If the total annual income from first two investments is Rs. 70 more than the income from the third, find the amount of each investment using matrix method.

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

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

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

…

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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