Matrices of order `3xx3` are formed by using the elements of the set `A={-3,-2,-1,0,1,2,3}`, then probability that matrix is either symmetric or skew symmetric is

Options

A. `(1)/(7^(6))+(1)/(7^(3))`
B. `(1)/(7^(9))+(1)/(7^(3))-(1)/(7^(6))`
C. `(1)/(7^(3))+(1)/(7^(9))`
D. `(1)/(7^(3))+(1)/(7^(6))-(1)/(7^(9))`

Explanation

To solve the problem of finding the probability that a randomly formed \(3 \times 3\) matrix using elements from the set \(A = \{-3, -2, -1, 0, 1, 2, 3\}\) is either symmetric or skew-symmetric, we will follow these steps:

### Step 1: Determine the total number of \(3 \times 3\) matrices
A \(3 \times 3\) matrix has \(9\) elements. Since each element can be chosen from the set \(A\) which contains \(7\) elements, the total number of \(3 \times 3\) matrices is given by:
\[ \text{Total matrices} = 7^9 \]

### Step 2: Calculate the number of symmetric matrices
A symmetric matrix satisfies the condition \(A^T = A\). For a \(3 \times 3\) symmetric matrix, the elements are arranged as follows:

\[ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{pmatrix} \]

Here, the diagonal elements \(a_{11}, a_{22}, a_{33}\) can be chosen freely from the set \(A\) (3 elements), and the off-diagonal elements \(a_{12}, a_{13}, a_{23}\) can also be chosen independently (3 elements). Therefore, the total number of symmetric matrices is:
\[ \text{Number of symmetric matrices} = 7^6 \]

### Step 3: Calculate the number of skew-symmetric matrices
A skew-symmetric matrix satisfies the condition \(A^T = -A\). For a \(3 \times 3\) skew-symmetric matrix, the elements are arranged as follows:

\[ \begin{pmatrix} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{pmatrix} \]

In this case, the diagonal elements must be \(0\). The off-diagonal elements can be chosen freely from the set \(A\) (3 elements). Thus, the total number of skew-symmetric matrices is:
\[ \text{Number of skew-symmetric matrices} = 7^3 \]

### Step 4: Find the intersection of symmetric and skew-symmetric matrices
The only matrix that is both symmetric and skew-symmetric is the zero matrix. Therefore, there is exactly \(1\) matrix that is both.

### Step 5: Use the principle of inclusion-exclusion
To find the total number of matrices that are either symmetric or skew-symmetric, we can use the formula:
\[ \text{Number of symmetric or skew-symmetric matrices} = \text{Number of symmetric matrices} + \text{Number of skew-symmetric matrices} - \text{Number of both} \]
Substituting the values we calculated:
\[ \text{Number of symmetric or skew-symmetric matrices} = 7^6 + 7^3 - 1 \]

### Step 6: Calculate the probability
The probability that a randomly formed matrix is either symmetric or skew-symmetric is given by:
\[ P = \frac{\text{Number of symmetric or skew-symmetric matrices}}{\text{Total matrices}} = \frac{7^6 + 7^3 - 1}{7^9} \]

### Step 7: Simplify the probability
We can simplify the probability:
\[ P = \frac{7^6 + 7^3 - 1}{7^9} = \frac{1}{7^3} + \frac{1}{7^6} - \frac{1}{7^9} \]

### Final Answer
Thus, the probability that a randomly formed \(3 \times 3\) matrix is either symmetric or skew-symmetric is:
\[ P = \frac{7^6 + 7^3 - 1}{7^9} \]