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 find the probability that a randomly formed \(3 \times 3\) matrix is skew-symmetric using the numbers from the box, we need to follow these steps:

### Step 1: Understanding Skew-Symmetric Matrices
A matrix \(A\) is skew-symmetric if \(A^T = -A\). For a \(3 \times 3\) matrix, this means:
\[ A = \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} \]
where \(a\), \(b\), and \(c\) are elements chosen from the box.

### Step 2: Identifying the Elements
The box contains the numbers: \(-3, -2, -1, 0, 1, 2, 3, 4, 5\). For a skew-symmetric matrix, we need to choose \(a\), \(b\), and \(c\) such that their negative counterparts \(-a\), \(-b\), and \(-c\) are also available in the box.

### Step 3: Choosing Elements
The valid choices for \(a\), \(b\), and \(c\) must be from the set of numbers that have corresponding negative values in the box. The valid pairs are:
- \(a\) can be from \{-3, -2, -1, 0, 1, 2, 3\} (7 choices)
- \(b\) can be from \{-3, -2, -1, 0, 1, 2, 3\} (7 choices)
- \(c\) can be from \{-3, -2, -1, 0, 1, 2, 3\} (7 choices)

### Step 4: Total Combinations
Since we are drawing numbers 9 times with replacement, the total number of possible outcomes for the \(3 \times 3\) matrix is:
\[ 9^9 \]
This is because each of the 9 positions in the matrix can be filled with any of the 9 numbers.

### Step 5: Favorable Outcomes
The favorable outcomes for forming a skew-symmetric matrix are determined by the choices for \(a\), \(b\), and \(c\):
- We have 7 choices for \(a\)
- We have 7 choices for \(b\)
- We have 7 choices for \(c\)

Thus, the total number of favorable outcomes is:
\[ 7 \times 7 \times 7 = 343 \]

### Step 6: Calculating the Probability
The probability \(P\) that a randomly formed \(3 \times 3\) matrix is skew-symmetric is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{343}{9^9} \]

### Final Answer
Thus, the probability that the matrix is skew-symmetric is:
\[ \frac{343}{387420489} \]