An amount of ₹ 10,000 is put into three investments at the rate of 10 %, 12 % and 15 % per annum. The combined annual income of all three investments…

Let $\displaystyle \mathrm{x}, \mathrm{y}$ and z (in ₹) be three investment amounts. Then $\displaystyle \boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{z}=\mathbf{1 0 , 0 0 0}$ \[\frac{10}{100} \mathrm{x}+\frac{12}{100} \mathrm{y}+\frac{15}{100} z=1310 \] Let $\displaystyle A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 10 & 12 & 15 \\ -10 & -12 & 15 \end{array}\right], X=\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ z\end{array}\right], B=\left[\begin{array}{c}10,000 \\ 1,31,000 \\ 19,000\end{array}\right]$ \[|\mathbf{A}|=\mathbf{6 0} \neq \mathbf{0} \] $\displaystyle \therefore \mathbf{A}^{-\mathbf{1}}$ exists