Question: Determine whether the statement below is true or false. Justify the answer.
If $A$ is an invertible $n \times n$ matrix, then the equation $A\mathbf{x} = \mathbf{b}$ is consistent for each $\mathbf{b}$ in $\mathbb{R}^n$.
Choose the correct answer below.
A. The statement is false. The matrix $A$ satisfies $A\mathbf{x} = \mathbf{b}$ if and only if $A$ is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible.
B. The statement is true. Since $A$ is invertible, $A^{-1}\mathbf{b}$ exists for all $\mathbf{b}$ in $\mathbb{R}^n$. Define $\mathbf{x} = A^{-1}\mathbf{b}$. Then $A\mathbf{x} = \mathbf{b}$.
C. The statement is true. Since $A$ is invertible, $A^{-1}\mathbf{b} = \mathbf{x}$ for all $\mathbf{x}$ in $\mathbb{R}^n$. Multiply both sides by $A$ and the result is $A\mathbf{x} = \mathbf{b}$.
D. The statement is false. The matrix $A$ is invertible if and only if $A$ is row equivalent to the identity matrix, and not every matrix $A$ satisfying $A\mathbf{x} = \mathbf{b}$ is row equivalent to the identity matrix.
1. **State the problem:** Determine if the statement "If $A$ is an invertible $n \times n$ matrix, then the equation $A\mathbf{x} = \mathbf{b}$ is consistent for each $\mathbf{b}$ in $\mathbb{R}^n$" is true or false.
2. **Recall the definition of invertibility:** A matrix $A$ is invertible if there exists a matrix $A^{-1}$ such that
$$A^{-1}A = AA^{-1} = I,$$
where $I$ is the identity matrix.
3. **Implication of invertibility on solutions:** If $A$ is invertible, then for any vector $\mathbf{b}$ in $\mathbb{R}^n$, the equation
$$A\mathbf{x} = \mathbf{b}$$
has a unique solution given by
$$\mathbf{x} = A^{-1}\mathbf{b}.$$
4. **Consistency of the equation:** Since $A^{-1}$ exists, $A^{-1}\mathbf{b}$ is defined for every $\mathbf{b}$ in $\mathbb{R}^n$, so the equation $A\mathbf{x} = \mathbf{b}$ is consistent for all $\mathbf{b}$.
5. **Check the options:**
- Option A is false because every invertible matrix is row equivalent to the identity matrix, and the statement about row equivalence not implying invertibility is incorrect here.
- Option B correctly states that since $A$ is invertible, $A^{-1}\mathbf{b}$ exists for all $\mathbf{b}$ and defines $\mathbf{x} = A^{-1}\mathbf{b}$, so $A\mathbf{x} = \mathbf{b}$.
- Option C is similar to B but incorrectly states "for all $\mathbf{x}$" instead of "for all $\mathbf{b}$".
- Option D is false because it incorrectly claims not every matrix satisfying $A\mathbf{x} = \mathbf{b}$ is row equivalent to the identity matrix, which contradicts invertibility.
**Final answer:** The correct choice is **B**.