If A and B are square matrices of size `nxxn` such that `A^(2)-B^(2)=(A-B)(A+B)`, then which of the following will be always true

Options

A. AB=BA
B. A=B
C. A or B=O
D. A or B=I

Answer

To solve the problem, we start with the given equation involving square matrices \( A \) and \( B \):

\[ A^2 - B^2 = (A - B)(A + B) \]

### Step 1: Understand the Equation
The equation \( A^2 - B^2 = (A - B)(A + B) \) is a well-known identity in algebra that holds for real numbers. We need to verify if it holds for matrices and what conditions can be derived from it.

### Step 2: Expand the Right Side
We can expand the right side of the equation:

\[ (A - B)(A + B) = A^2 + AB - BA - B^2 \]

### Step 3: Set the Expanded Form Equal to the Left Side
Now, we set the expanded form equal to the left side of the original equation:

\[ A^2 - B^2 = A^2 + AB - BA - B^2 \]

### Step 4: Cancel Out Common Terms
By cancelling \( A^2 \) and \( -B^2 \) from both sides, we have:

\[ 0 = AB - BA \]

### Step 5: Rearranging the Equation
This simplifies to:

\[ AB = BA \]

### Conclusion
This indicates that matrices \( A \) and \( B \) commute with each other. Therefore, the condition that must always be true is:

\[ AB = BA \]