Find the Inverse, If It Exists, for the Matrix $$\left[ \begin{array} { r r r r } 0 & 5 & 0 & - 3 \\4 & 15 & - 3 & - 6 \\8 & 10 & - 6 & - 3 \\12 & 0 & - 6 & 0\end{array} \right]$$

question 366

Multiple Choice

Find the inverse, if it exists, for the matrix.
- $$\left[ \begin{array} { r r r r } 0 & 5 & 0 & - 3 \\4 & 15 & - 3 & - 6 \\8 & 10 & - 6 & - 3 \\12 & 0 & - 6 & 0\end{array} \right]$$

• A)
• \[\left[ \begin{array} { r r r r }
• \frac { 1 } { 2 } & - \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 4 } \\
• \frac { 3 } { 5 } & - \frac { 2 } { 5 } & \frac { 1 } { 5 } & 0 \\
• 1 & - 1 & 1 & \frac { 1 } { 3 } \\
• - 1 & - \frac { 2 } { 3 } & \frac { 1 } { 3 } & 0
• \end{array} \right]\]
• B) The inverse does not exist.
• C)
• \[\left[ \begin{array} { r r r r }
• - \frac { 1 } { 2 } & - \frac { 3 } { 5 } & - 1 & - \frac { 4 } { 3 } \\
• \frac { 1 } { 2 } & \frac { 2 } { 5 } & 1 & \frac { 1 } { 3 } \\
• - \frac { 1 } { 2 } & - \frac { 1 } { 5 } & - 1 & - \frac { 1 } { 3 } \\
• \frac { 1 } { 4 } & 0 & \frac { 1 } { 3 } & 0
• \end{array} \right]\]
• D)
• \[\left[ \begin{array} { c c c c }
• - \frac { 1 } { 2 } & \frac { 1 } { 2 } & - \frac { 1 } { 2 } & \frac { 1 } { 4 } \\
• - \frac { 3 } { 5 } & \frac { 2 } { 5 } & - \frac { 1 } { 5 } & 0 \\
• - 1 & 1 & - 1 & \frac { 1 } { 3 } \\
• - \frac { 4 } { 3 } & \frac { 2 } { 3 } & - \frac { 1 } { 3 } & 0
• \end{array} \right]\]

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Definitions:

Short Run

A period in economics during which some factors of production or inputs are fixed in quantity and cannot be changed.

Industry

A distinct group of productive or service businesses that share similar characteristics, processes, or goods.

Perfect Competitor

A theoretical firm in a perfectly competitive market where no single buyer or seller has the market power to influence prices.

Long Run

The long run is a period in which all factors of production and costs are variable, allowing for complete adjustment to changes.