The number of `nxxn` matrix A and B such that AB - BA = I is. . .

Options

A. infinite
B. `n^(2)`
C. nl
D. zero

Answer

To solve the problem of finding the number of \( n \times n \) matrices \( A \) and \( B \) such that \( AB - BA = I \), we can follow these steps:

### Step 1: Understand the equation \( AB - BA = I \)

We start with the equation \( AB - BA = I \), where \( I \) is the identity matrix of size \( n \times n \). This equation suggests that the commutator of matrices \( A \) and \( B \) results in the identity matrix.

### Step 2: Analyze the trace of both sides

The trace of a matrix is the sum of its diagonal elements. A key property of the trace is that it is linear and that the trace of the commutator of two matrices is always zero:

\[ \text{trace}(AB - BA) = \text{trace}(AB) - \text{trace}(BA) = 0 \]

Thus, we have:

\[ \text{trace}(I) = n \quad \text{(for an \( n \times n \) identity matrix)} \]

### Step 3: Equate the traces

Since we found that \( \text{trace}(AB - BA) = 0 \) and \( \text{trace}(I) = n \), we can equate these:

\[ 0 \neq n \]

This indicates that there is a contradiction because the left-hand side (trace of the commutator) cannot equal the right-hand side (trace of the identity matrix) for any \( n \).

### Step 4: Conclusion

Since the traces are not equal, it implies that there are no matrices \( A \) and \( B \) such that \( AB - BA = I \). Therefore, the number of such matrices is:

\[ \text{Number of matrices } A \text{ and } B = 0 \]

### Final Answer

The number of \( n \times n \) matrices \( A \) and \( B \) such that \( AB - BA = I \) is \( 0 \).

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