Discover Linear Algebra: A first course in linear algebra

Summarize the main idea(s) of the reasoning behind each of the statements in Proposition 9.4.2.

Explain in your own words why a square matrix and its transpose always have the same determinant value.

State the determinant pattern for each of the three types of elementary matrices, and the reasoning for that pattern based on a particular statement from Proposition 9.4.2.

True or false: For every square matrix $A$ and every scalar value $k\text{,}$ it is always true that $det(kA)=kdetA\text{.}$

If the statement is true, explain the reasoning behind it; if it is false, state the correct relationship and explain the reasoning behind that relationship.