A box contains 9 slips bearing numbers `-3, -2, -1, 0, 1, 2, 3, 4 and 5`. An experiment consists of drawing a slip from this box and replacing it back in the box after noting the number. This experiment is repeated 9 times. This experiment is repeaed 9 times. These 9 numbers are now chosen as elements of `3xx3` matrix, then the probability that the matrix is skew symmetric is

Options

A. `(1)/(9^(6))`
B. `(343)/(9^(9))`
C. `(1)/(9^(9))`
D. `(1)/(9^(7))`

Answer

To solve the problem of finding the probability that a randomly formed \(3 \times 3\) matrix is skew-symmetric, we can follow these steps:

### Step 1: Understand the conditions for a skew-symmetric matrix
A \(3 \times 3\) matrix \(A\) is skew-symmetric if \(A^T = -A\). This means that:
- The diagonal elements must be zero: \(a_{11} = a_{22} = a_{33} = 0\).
- The off-diagonal elements must satisfy \(a_{ij} = -a_{ji}\) for \(i \neq j\).

### Step 2: Define the matrix structure
A general \(3 \times 3\) skew-symmetric matrix can be represented as:
\[ A = \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} \]
Here, \(a\), \(b\), and \(c\) are the off-diagonal elements.

### Step 3: Determine the number of choices for \(a\), \(b\), and \(c\)
The slips in the box contain the numbers \(-3, -2, -1, 0, 1, 2, 3, 4, 5\). Therefore, there are 9 possible values for each of \(a\), \(b\), and \(c\).

### Step 4: Calculate the total number of matrices
Since we are drawing 9 slips and forming a \(3 \times 3\) matrix, the total number of ways to choose the 9 numbers (with replacement) is:
\[ 9^9 \]

### Step 5: Calculate the number of favorable outcomes for skew-symmetric matrices
- The diagonal elements must be 0, which has 1 way to choose (only the number 0).
- For \(a\), \(b\), and \(c\), each can be any of the 9 numbers. Thus, there are \(9\) choices for \(a\), \(9\) choices for \(b\), and \(9\) choices for \(c\).

The total number of favorable outcomes for forming a skew-symmetric matrix is:
\[ 9 \times 9 \times 9 = 9^3 \]

### Step 6: Calculate the probability
The probability \(P\) that a randomly formed \(3 \times 3\) matrix is skew-symmetric is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9^3}{9^9} = \frac{1}{9^6} \]

### Final Result
Thus, the probability that the matrix is skew-symmetric is:
\[ \frac{1}{9^6} \]