If A and B are two square matrices such that AB=BA then `(A-B)^(2) =A^(2) - 2AB + B^(2)`. True or False.

Answer

To determine whether the statement \((A - B)^2 = A^2 - 2AB + B^2\) is true or false given that \(AB = BA\), we can follow these steps:

### Step 1: Expand \((A - B)^2\)
We start with the expression \((A - B)^2\):
\[ (A - B)^2 = (A - B)(A - B) \]

### Step 2: Apply the distributive property
Using the distributive property (also known as the FOIL method for binomials), we can expand this:
\[ (A - B)(A - B) = A \cdot A - A \cdot B - B \cdot A + B \cdot B \]
This simplifies to:
\[ = A^2 - AB - BA + B^2 \]

### Step 3: Use the commutativity property
Since we are given that \(AB = BA\), we can replace \(BA\) with \(AB\):
\[ = A^2 - AB - AB + B^2 \]
This simplifies to:
\[ = A^2 - 2AB + B^2 \]

### Conclusion
Thus, we have shown that:
\[ (A - B)^2 = A^2 - 2AB + B^2 \]
Therefore, the statement is **True**.