Under what condition does A (BC) = (AB)C hold, where A, B, C are three matrices ?

Options

A. AB and BC both must exist
B. Only Ab must exist
C. Only BC must exist
D. Always true

Explanation

To determine the condition under which the equation \( A(BC) = (AB)C \) holds for three matrices \( A \), \( B \), and \( C \), we need to explore the properties of matrix multiplication, specifically the associative property.

### Step-by-Step Solution:

1. **Understanding Matrix Multiplication**:
- Matrix multiplication is not commutative, meaning \( AB \neq BA \) in general. However, it is associative, which means that the way in which matrices are grouped during multiplication does not change the result, provided the matrices are compatible for multiplication.

2. **Identifying Compatibility**:
- For the expression \( A(BC) \) to be valid, the matrices \( B \) and \( C \) must be compatible for multiplication. This means that the number of columns in \( B \) must equal the number of rows in \( C \).
- For the expression \( (AB)C \) to be valid, the matrices \( A \) and \( B \) must also be compatible for multiplication, which means the number of columns in \( A \) must equal the number of rows in \( B \).

3. **Conditions for Equality**:
- The equation \( A(BC) = (AB)C \) holds true if both products \( AB \) and \( BC \) are defined. Therefore, we need to ensure that:
- \( A \) has the same number of columns as \( B \) has rows.
- \( B \) has the same number of columns as \( C \) has rows.

4. **Conclusion**:
- The condition under which \( A(BC) = (AB)C \) holds is that both products \( AB \) and \( BC \) must exist. This means that:
- \( A \) must be compatible with \( B \) (i.e., the number of columns in \( A \) equals the number of rows in \( B \)).
- \( B \) must be compatible with \( C \) (i.e., the number of columns in \( B \) equals the number of rows in \( C \)).

### Final Condition:
The condition under which \( A(BC) = (AB)C \) holds is:
- Both \( AB \) and \( BC \) must exist.