Find the Matrix of the Quadratic Form $$4 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }$$

question 11

Multiple Choice

Find the matrix of the quadratic form.
- $$4 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }$$

• A)
• \(\left[ \begin{array} { c r c } 4 & - 2 & 1 / 2 \\ - 2 & 5 & 0 \\ 1 / 2 & 0 & 3 \end{array} \right]\)
• B)
• \(\left[ \begin{array} { r r r } 4 & - 2 & 0 \\ - 2 & 5 & 1 / 2 \\ 0 & 1 / 2 & 3 \end{array} \right]\)
• C)
• \(\left[ \begin{array} { r r r } 4 & - 4 & 0 \\ - 4 & 5 & 1 \\ 0 & 1 & 3 \end{array} \right]\)
• D)
• \(\left[ \begin{array} { l l l } 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 3 \end{array} \right]\)

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Definitions:

Cost of Equity

The rate of return that a company is expected to pay to its shareholders to compensate them for the risk of investing in the equity of the company.

Leverage

The use of various financial instruments or borrowed capital, like debt, to increase the potential return of an investment.

Debt-Equity Ratio

The ratio of a firm's total liabilities to its shareholders' equity, used to assess financial leverage.

Financial Risk

The possibility of losing money on an investment or business venture, often associated with the uncertainty of returns.