Question

The number of symmetric matrices of order 3, with all the entries from the set { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } is

Answer 1

10^6

Answer 2

A 3x3 symmetric matrix $A$ is defined by the property $a_{ij} = a_{ji}$ for all $i, j$. This means that the elements below the main diagonal are determined by the elements above the main diagonal. For a 3x3 matrix:

$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$

The symmetry condition implies $a_{21} = a_{12}$, $a_{31} = a_{13}$, and $a_{32} = a_{23}$. The independent entries that can be chosen freely are the diagonal elements ($a_{11}, a_{22}, a_{33}$) and the elements in the upper triangle ($a_{12}, a_{13}, a_{23}$). In total, there are $3 + 3 = 6$ independent entries.

The set of allowed entries is $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, which contains 10 distinct values. Since each of the 6 independent entries can be chosen in 10 ways, the total number of distinct symmetric matrices is the product of the number of choices for each independent entry: Total number of matrices = $10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^6$.

This can be generalized for an $n \times n$ symmetric matrix with entries from a set of size $m$. The number of independent entries is $\frac{n(n+1)}{2}$. The total number of such matrices is $m^{\frac{n(n+1)}{2}}$. For $n=3$ and $m=10$, this is $10^{\frac{3(3+1)}{2}} = 10^6$.