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Reflections 10.7 Reflect on your understanding
In addition to the reflection activities below, re-read
Section 10.2 Terminology and notation . Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
1. The role of the adjoint matrix.
Consider the formula \(A (\adj A) = (\det A) I\text{.}\)
(a) Briefly explain why the formula is true for all square matrices. (Do not assume the size of the matrix — your explanation should be equally valid for all sizes of square matrices.)
(b) Briefly explain the relationship between this formula and the concept of matrix inverse .
2. Determinants of row equivalent matrices.
Explain why the determinants of two row equivalent matrices must be either both zero or both nonzero.
3. Cramer’s rule.
Briefly summarize Cramer’s rule and how it follows from the adjoint inversion formula.
