Let I denote the `3xx3` identity matrix and P be a matrix obtained by rearranging the columns of I. then

Options

A. there are six distinct choices for P and det (P) = 1
B. there are six distinct choices for P and det (P) = `pm` 1
C. there are more than one choice for P and some of them are not invertible
D. there are more than one choice for P and `P^(-1)` = I in each choice

Answer

To solve the problem, we need to analyze the properties of the identity matrix and the effects of rearranging its columns. Let's go through the steps systematically.

### Step 1: Define the Identity Matrix
The 3x3 identity matrix \( I \) is defined as:
\[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Understand Rearranging Columns
The matrix \( P \) is obtained by rearranging the columns of the identity matrix \( I \). Rearranging columns means that we can permute the columns of \( I \) in any order.

### Step 3: Count Distinct Rearrangements
Since there are 3 columns in the identity matrix, the number of distinct ways to rearrange these columns is given by the factorial of the number of columns:
\[ 3! = 6 \]
Thus, there are 6 distinct matrices \( P \) that can be formed by rearranging the columns of \( I \).

### Step 4: Determine the Determinant of \( P \)
The determinant of the identity matrix \( I \) is:
\[ \text{det}(I) = 1 \]
When we rearrange the columns of a matrix, the determinant can either remain the same or change sign depending on whether the number of swaps (exchanges) is even or odd.

- If we perform an even number of swaps, the determinant remains \( +1 \).
- If we perform an odd number of swaps, the determinant becomes \( -1 \).

Thus, for the matrix \( P \), the determinant can take values:
\[ \text{det}(P) = \pm 1 \]

### Conclusion
In conclusion, the determinant of the matrix \( P \) obtained by rearranging the columns of the identity matrix \( I \) can be either \( +1 \) or \( -1 \).

### Final Answer
The determinant of \( P \) is \( \pm 1 \).

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