Perform the Row Operation(s) on the Given Augmented Matrix $R _ { 2 } = - 2 r _ { 1 } + r _ { 2 }$

question 201

Multiple Choice

Perform the row operation(s) on the given augmented matrix.
-(a) $R _ { 2 } = - 2 r _ { 1 } + r _ { 2 }$
(b) $\mathrm { R } _ { 3 } = - 2 \mathrm { r } _ { 1 } + \mathrm { r } _ { 3 }$
(c) $\mathrm { R } _ { 3 } = 6 \mathrm { r } _ { 2 } + \mathrm { r } _ { 3 }$
$$\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\2 & - 5 & - 4 & 5 \\2 & 5 & 4 & 6\end{array} \right]$$

• A) \(\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & 4 & 11 & 11 \\ 0 & 21 & 35 & 25 \end{array} \right]\)
• B) \(\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & 4 & 11 & 11 \\ 0 & 37 & 79 & 69 \end{array} \right]\)
• C) \(\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & - 8 & - 9 & 3 \\ 0 & - 37 & - 40 & - 8 \end{array} \right]\)
• D) \(\left[ \begin{array} { r r r | r } 1 & - 3 & - 5 & - 2 \\ 0 & 14 & 19 & 4 \\ 0 & 87 & 119 & 34 \end{array} \right]\)

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Definitions:

Allocative Efficiency

A state of resource allocation where it is impossible to make any one individual better off without making someone else worse off, effectively maximizing overall social welfare.

Productive Efficiency

A situation where an economy or entity cannot produce more of one good without reducing the output of another good, indicating optimal resource allocation.

Monopoly Power

Monopoly power refers to the ability of a firm to control market prices and production, typically due to lack of competition in the marketplace.

Product Variety

The assortment of different products or services offered by a firm or available in a market.