Given the Following Matrices Show That $( \boldsymbol { A } + \boldsymbol { B } ) ^ { \prime } = \boldsymbol { A } ^ { \prime } + \boldsymbol { B } ^ { \prime }$

question 9

Essay

Given the following matrices $$\boldsymbol { A } = \left( \begin{array} { l l } a _ { 11 } & a _ { 12 } \\a _ { 21 } & a _ { 22 }\end{array} \right) , \boldsymbol { B } = \left( \begin{array} { l l } b _ { 11 } & b _ { 12 } \\b _ { 21 } & b _ { 22 }\end{array} \right) , \text { and } \boldsymbol { C } = \left( \begin{array} { l l l } c _ { 11 } & c _ { 12 } & c _ { 13 } \\c _ { 21 } & c _ { 22 } & c _ { 23 }\end{array} \right)$$
show that $( \boldsymbol { A } + \boldsymbol { B } ) ^ { \prime } = \boldsymbol { A } ^ { \prime } + \boldsymbol { B } ^ { \prime }$ and $( \boldsymbol { A } \boldsymbol { C } ) ^ { \prime } = \boldsymbol { C } ^ { \prime } \boldsymbol { A } ^ { \prime }$ .

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Definitions:

Agent Liability

Refers to the legal responsibility of an agent to their principal, including obligations and potential for repercussions if the agent acts outside their authority or against the interests of the principal.

Principal

In law, the person who authorizes an agent to act on their behalf; in finance, the original amount of money invested or loaned.

Vicarious Liability

The liability or responsibility imposed on a person, a party, or an organization for damages caused by another; most commonly used in relation to employment, with the employer held vicariously liable for damages caused by its employees.

Principal-Employer

The party in a contractual relationship who has the right to control the work of the employee or agent and is responsible for their actions.