1. **Stating the problem:** Prove the equivalence of the statements in Theorem 1.6.4 about an $n \times n$ matrix $A$: (a) $A$ is invertible. (b) $Ax = 0$ has only the trivial solution. (c) The reduced row echelon form of $A$ is $I_n$. (d) $A$ is expressible as a product of elementary matrices. (e) $Ax = b$ is consistent for every $n \times 1$ matrix $b$. (f) $Ax = b$ has exactly one solution for every $n \times 1$ matrix $b$. 2. **Known equivalences:** From Theorem 1.5.3, (a), (b), (c), and (d) are equivalent. 3. **Prove (a) $\Rightarrow$ (f):** If $A$ is invertible, then for every $b$, the equation $Ax = b$ has a unique solution given by $$x = A^{-1}b$$ This shows (f) holds. 4. **Prove (f) $\Rightarrow$ (e):** If $Ax = b$ has exactly one solution for every $b$, then it is certainly consistent for every $b$, so (e) holds. 5. **Prove (e) $\Rightarrow$ (a):** If $Ax = b$ is consistent for every $b$, then the columns of $A$ span $\mathbb{R}^n$, implying $A$ is invertible. 6. **Summary:** Since (a), (b), (c), (d) are equivalent and (a) $\Rightarrow$ (f) $\Rightarrow$ (e) $\Rightarrow$ (a), all statements (a) through (f) are equivalent. **Final answer:** The statements (a) through (f) in Theorem 1.6.4 are equivalent.