Use the Appropriate Property of Determinants from This Section to Justify

question 198

Multiple Choice

Use the appropriate property of determinants from this section to justify the true statement. $$\left| \begin{array} { c c c } 12 & 3 & 3 \\4 & - 1 & 9 \\20 & - 5 & 15\end{array} \right| = 12 \left| \begin{array} { c c c } 3 & 3 & 1 \\1 & - 1 & 3 \\5 & - 5 & 5\end{array} \right| = 60 \left| \begin{array} { c c c } 3 & 3 & 1 \\1 & - 1 & 3 \\1 & - 1 & 1\end{array} \right|$$ Do not evaluate the determinants.

• A) Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |.
• B) Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = \(-\) | A |.
• C) Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |.
• D) Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0.
• E) Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0.

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Definitions:

Madison

James Madison, the fourth President of the United States, known as the "Father of the Constitution" for his pivotal role in drafting and promoting the U.S. Constitution and the Bill of Rights.

Anti-Federalists

Individuals who opposed the ratification of the U.S. Constitution in 1787-1788, fearing it centralized too much power in the federal government and outside the states.

Human Nature

The inherent characteristics, including ways of thinking, feeling, and acting, that all human beings share, despite cultural or individual differences.

Republican Institutions

Governance structures and practices based on the principles of republicanism, where the government's authority comes from the people and is exercised by elected representatives.