From the matrix equation AB=AC we can conclude B=C provided the matrix A is

Options

A. singular
B. non singular
C. symmetric
D. None of these

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. **Rearrange the equation**:
We can rewrite the equation as:
\[ AB - AC = 0 \]
This can be factored as:
\[ A(B - C) = 0 \]

3. **Consider the properties of matrix A**:
To conclude that \( B = C \), we need to consider the properties of matrix \( A \). Specifically, we need to check if \( A \) is non-singular (invertible). A matrix is non-singular if its determinant is not equal to zero:
\[ \text{det}(A) \neq 0 \]

4. **Multiply both sides by the inverse of A**:
If \( A \) is non-singular, we can multiply both sides of the equation \( A(B - C) = 0 \) by \( A^{-1} \):
\[ A^{-1}(A(B - C)) = A^{-1}0 \]
This simplifies to:
\[ I(B - C) = 0 \]
where \( I \) is the identity matrix.

5. **Conclude that \( B - C = 0 \)**:
From the equation \( I(B - C) = 0 \), we can conclude:
\[ B - C = 0 \implies B = C \]

6. **Final conclusion**:
Therefore, we can conclude that \( B = C \) provided that the matrix \( A \) is non-singular.

### Summary:
From the matrix equation \( AB = AC \), we can conclude \( B = C \) if \( A \) is non-singular (invertible).