From the matrix equation AB=AC, we conclude B=C provided.

Options

A. A is singular
B. A is non-singular
C. A is symmetric
D. A is a square

Answer

To solve the problem, we need to analyze the matrix equation \( AB = AC \) and determine under what conditions we can conclude that \( B = C \).

### Step-by-step Solution:

1. **Start with the given equation**:
\[ AB = AC \]

2. **Subtract \( AC \) from both sides**:
\[ AB - AC = 0 \]

3. **Factor out \( A \) from the left-hand side**:
\[ A(B - C) = 0 \]

4. **Analyze the product \( A(B - C) = 0 \)**:
This equation states that the product of matrix \( A \) and the matrix \( (B - C) \) equals the zero matrix.

5. **Consider the implications of this product being zero**:
For the equation \( A(B - C) = 0 \) to imply \( B - C = 0 \) (or \( B = C \)), \( A \) must be a non-singular matrix. A non-singular matrix is one that has an inverse, which means its determinant is non-zero.

6. **Multiply both sides by \( A^{-1} \)** (the inverse of \( A \)):
\[ A^{-1}(A(B - C)) = A^{-1}0 \]
This simplifies to:
\[ I(B - C) = 0 \]
where \( I \) is the identity matrix.

7. **Conclude that**:
\[ B - C = 0 \quad \Rightarrow \quad B = C \]

### Conclusion:
Thus, we conclude that \( B = C \) provided that \( A \) is a non-singular matrix (i.e., \( \text{det}(A) \neq 0 \)).