Matrices A and B will be inverse of each other only if

Options

A. ` AB = BA `
B. ` AB = BA = 0 `
C. ` AB = 0, BA = I `
D. ` AB = BA = I `

Answer

To determine the conditions under which matrices A and B are inverses of each other, we need to establish the mathematical definitions and properties of matrix inverses.

### Step-by-Step Solution:

1. **Definition of Inverse Matrices**:
- Two matrices A and B are said to be inverses of each other if:
\[ AB = I \quad \text{and} \quad BA = I \]
where I is the identity matrix.

2. **Establishing the First Condition**:
- We start by multiplying matrix A by matrix B:
\[ AB = I \]
This means that when A is multiplied by B, the result is the identity matrix.

3. **Establishing the Second Condition**:
- Next, we multiply matrix B by matrix A:
\[ BA = I \]
This indicates that when B is multiplied by A, the result is also the identity matrix.

4. **Conclusion**:
- For matrices A and B to be inverses of each other, both conditions must hold true:
\[ AB = I \quad \text{and} \quad BA = I \]
Therefore, the correct conclusion is that matrices A and B will be inverses of each other if both conditions are satisfied.