Suppose A ,B are two `3xx3` matrices such that `A^(-1)` exists . Then `(A-B) A^(-1) (A+B) ` is equal to

Options

A. `(A+B)(A^(-1)) (A-B) `
B. `A^(-1)B+B^(2)`
C. `(I-BAB^(-1)) (A-B)`
D. `(I+BAB^(-1))(A+B)`

Answer

To solve the problem, we need to simplify the expression \((A - B) A^{-1} (A + B)\).

### Step 1: Rewrite the expression
We start with the expression:
\[ (A - B) A^{-1} (A + B) \]

### Step 2: Distribute \(A^{-1}\)
We can distribute \(A^{-1}\) into \((A + B)\):
\[ = (A - B) (A^{-1} A + A^{-1} B) \]
Since \(A^{-1} A\) is the identity matrix \(I\), we have:
\[ = (A - B) (I + A^{-1} B) \]

### Step 3: Distribute \((A - B)\)
Now, we distribute \((A - B)\):
\[ = (A - B) I + (A - B) A^{-1} B \]
This simplifies to:
\[ = A - B + (A - B) A^{-1} B \]

### Step 4: Simplify further
Now we can factor out \((A - B)\):
\[ = A - B + (A - B) A^{-1} B \]
This can be rewritten as:
\[ = (A - B)(I + A^{-1} B) \]

### Final Result
Thus, the final simplified expression is:
\[ (A - B)(I + A^{-1} B) \]