Use the Appropriate Property of Determinants from This Section to Justify

question 148

Short Answer

Use the appropriate property of determinants from this section to justify the true statement. $$\left| \begin{array} { c c c } 7 & 9 & 15 \\8 & - 12 & 4 \\7 & 6 & - 1\end{array} \right| = - \left| \begin{array} { c c c } 7 & 15 & 9 \\8 & 4 & - 12 \\7 & - 1 & 6\end{array} \right|$$ Do not evaluate the determinants. Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0. Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | 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 |. 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 |. Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0. Enter property number only.

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Definitions:

Aquaculture Sites

Locations designated for the breeding, rearing, and harvesting of plants and animals in all types of water environments.

Great Lakes

A group of five large interconnected freshwater lakes in North America that serve as a natural border between the United States and Canada.

Ballast Waters

Water carried in ships' ballast tanks to improve stability and balance, which can introduce invasive species when discharged in new locations.

Great Lakes

A group of five large freshwater lakes located on the border between the United States and Canada, consisting of Lakes Superior, Michigan, Huron, Erie, and Ontario, which together form the largest group of freshwater lakes by area in the world.