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Reflections 9.6 Reflect on your understanding
1. Proof patterns.
Summarize the main idea(s) of the reasoning behind each of the statements in
Proposition 9.4.2 .
2. Determinant of a transpose.
Explain in your own words why a square matrix and its transpose always have the same determinant value.
3. Determinants of elementary matrices.
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 .
4. Determinants of scalar multiples.
True or false: For every square matrix \(A\) and every scalar value \(k\text{,}\) it is always true that \(\det (k A) = k \det A\text{.}\)
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.
