If A and B are two non-zero `n × n` matrices such that `A^2 + B = A^2 B`, then

Options

A. `A^2 = I` or `B = I`
B. `A^2 B = BA^2`
C. `AB = I`
D. `A^2 B = I`

Answer

To solve the problem, we start with the given equation involving two non-zero \( n \times n \) matrices \( A \) and \( B \):

\[ A^2 + B = A^2 B \]

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate terms involving \( B \):

\[ A^2 + B - A^2 B = 0 \]

### Step 2: Factoring Out \( A^2 \)
Next, we can factor out \( A^2 \) from the left-hand side:

\[ A^2 - A^2 B + B = 0 \]

This can be rewritten as:

\[ A^2 (I - B) + B = 0 \]

### Step 3: Isolating \( B \)
Now, we can isolate \( B \):

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

### Step 4: Rearranging Further
Rearranging gives us:

\[ A^2 I - A^2 B = -B \]

Adding \( B \) to both sides results in:

\[ A^2 I = A^2 B - B \]

### Step 5: Factoring Out \( B \)
We can factor \( B \) out on the right-hand side:

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

### Step 6: Analyzing the Equation
From the equation \( A^2 I = B (A^2 - I) \), we can see that if \( A^2 - I \) is invertible, we can multiply both sides by \( (A^2 - I)^{-1} \):

\[ B = A^2 I (A^2 - I)^{-1} \]

### Step 7: Conclusion
Now we can analyze the implications of this equation. We have established that:

\[ A^2 B = B A^2 \]

This indicates that \( A^2 \) and \( B \) commute.

### Final Result
Thus, we conclude that:

\[ A^2 B = B A^2 \]

This means that \( A^2 \) and \( B \) commute, which leads us to the conclusion that:

\[ A^2 B = B A^2 \]