The prices of three commodities P, Q and R are Rs x, y and z per unit respectively.

The prices of three commodities $P, Q$ and $R$ are Rs $x$, Rs $y$ and Rs $z$ per unit, respectively.

According to the question,

$3 x+5 y-4 z=6000$

$2 x-3 y+z=5000$

$-x+4 y+6 z=13000$

The given system of equations can be written in matrix form as follows:

$\left[\begin{array}{ccc}3 & 5 & -4 \\ 2 & -3 & 1 \\ -1 & 4 & 6\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}6000 \\ 5000 \\ 13000\end{array}\right]$

$A X=B$

Here,

$A=\left[\begin{array}{ccc}3 & 5 & -4 \\ 2 & -3 & 1 \\ -1 & 4 & 6\end{array}\right] X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] B=\left[\begin{array}{c}6000 \\ 5000 \\ 13000\end{array}\right]$

Now,

$|A|=3(-18-4)-5(12+1)-4(8-3)$

$=-66-65-20$

$=-151 \neq 0$

So, $A^{-1}$ exists.

Let $\mathrm{C}_{i j}$ be the cofactors of the elements $\mathrm{a}_{i j}$ in $\mathrm{A}=\left[a_{i j}\right]$. Then,

$C_{11}=(-1)^{1+1}\left|\begin{array}{cc}-3 & 1 \\ 4 & 6\end{array}\right|=-22, C_{12}=(-1)^{1+2}\left|\begin{array}{cc}2 & 1 \\ -1 & 6\end{array}\right|=-13, C_{13}=(-1)^{1+3}\left|\begin{array}{cc}2 & -3 \\ -1 & 4\end{array}\right|=5$

$C_{21}=(-1)^{2+1}\left|\begin{array}{cc}5 & -4 \\ 4 & 6\end{array}\right|=-46, C_{22}=(-1)^{2+2}\left|\begin{array}{cc}3 & -4 \\ -1 & 6\end{array}\right|=14, C_{23}=(-1)^{2+3}\left|\begin{array}{cc}3 & 5 \\ -1 & 4\end{array}\right|=-17$

$C_{31}=(-1)^{3+1}\left|\begin{array}{cc}5 & -4 \\ -3 & 1\end{array}\right|=-7, C_{32}=(-1)^{3+2}\left|\begin{array}{cc}3 & -4 \\ 2 & 1\end{array}\right|=-11, C_{33}=(-1)^{3+3}\left|\begin{array}{cc}3 & 5 \\ 2 & -3\end{array}\right|=-19$

adj $A=\left[\begin{array}{ccc}-22 & -13 & 5 \\ -46 & 14 & -17 \\ -7 & -11 & -19\end{array}\right]^{T}$

$=\left[\begin{array}{ccc}-22 & -46 & -7 \\ -13 & 14 & -11 \\ 5 & -17 & -19\end{array}\right]$

$\Rightarrow A^{-1}=\frac{1}{|A|} \operatorname{adj} A$

$=\frac{1}{-151}\left[\begin{array}{ccc}-22 & -46 & -7 \\ -13 & 14 & -11 \\ 5 & -17 & -19\end{array}\right]$

$X=A^{-1} B$

$\Rightarrow X=\frac{1}{{ }_{-151}}\left[\begin{array}{ccc}-22 & -46 & -7 \\ -13 & 14 & -11 \\ 5 & -17 & -19\end{array}\right]\left[\begin{array}{c}6000 \\ 5000 \\ 13000\end{array}\right]$

$\Rightarrow X=\frac{1}{-151}\left[\begin{array}{l}-453000 \\ -151000 \\ -302000\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}3000 \\ 1000 \\ 2000\end{array}\right]$

$\therefore x=3000, y=1000$ and $z=2000$

Thus, the prices of the three commodities $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are Rs 3000, Rs 1000 and Rs 2000 per unit, respectively.