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Reflections 6.7 Reflect on your understanding
In addition to the reflection activities below, re-read
Section 6.2 Terminology and notation . Be sure you understand the new definitions introduced in this chapter, and spend some time committing it to memory.
1. Elementary matrices.
Explain the role that elementary matrices play in matrix algebra.
2. Inverses of elementary matrices.
Why every elementary matrix is invertible.
Why the inverse of an elementary matrix is also elementary, of the same type.
What the inverse of an elementary matrix represents.
3. Linking elementary matrices to the theory of invertible matrices.
Briefly summarize the main idea(s) behind each step in the proof of
Theorem 6.5.2 .
4. Uniqueness of decompositions.
Statement 7 of
Theorem 6.5.2 says that every invertible matrix can be expressed as a product of elementary matrices. For a given invertible matrix, is there only
one unique such expression, or are multiple such expressions possible? Explain.
5. Revisiting row equivalence.
(a) (b) Use your response to
Task a to verify each of the following statements.
(i) If matrix \(B\) is row equivalent to matrix \(A\text{,}\) then there exists an invertible matrix \(P\) so that \(B = P A\text{.}\)
(ii) If there exists an invertible matrix \(P\) so that \(B = P A\text{,}\) then matrix \(B\) is row equivalent to matrix \(A\text{.}\)
(c) State the relationship between a matrix and its RREF as an algebraic equality involving the matrix, its RREF matrix, and elementary matrices.
(d) Explain why two square matrices of the same size that are row equivalent must either be both invertible or both singular.
