Here's a detailed and educational solution to the problem:

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Understanding the Key Concept: Symmetric Matrices

A symmetric matrix is a square matrix that is equal to its own transpose. Mathematically, for a matrix $A$ to be symmetric, it must satisfy $A = A^T$. This property implies a crucial relationship between its elements: for every element $A_{ij}$ (the element in row $i$ and column $j$), it must be equal to $A_{ji}$ (the element in row $j$ and column $i$). That is, $A_{ij} = A_{ji}$ for all $i$ and $j$.

Our goal is to find the number of $3 \times 3$ symmetric matrices where each entry can be chosen from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

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Step 1: Visualizing a $3 \times 3$ Symmetric Matrix

Let's consider a general $3 \times 3$ matrix:

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

Now, we apply the condition for a symmetric matrix, $A_{ij} = A_{ji}$:

• $A_{12} = A_{21}$
• $A_{13} = A_{31}$
• $A_{23} = A_{32}$

The elements on the main diagonal ($A_{11}, A_{22}, A_{33}$) are equal to themselves ($A_{ii} = A_{ii}$), so they don't impose any further restrictions based on symmetry.

Substituting these conditions into the general matrix structure, we can see which elements are independent choices and which are determined by others:

$$A = \begin{pmatrix} a & b & c \\ b & d & e \\ c & e & f \end{pmatrix}$$

Here, we've used different letters to represent the independent entries.

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Step 2: Identifying the Number of Independent Entries

From the structure above, we can clearly identify the entries that we are free to choose. Once these are chosen, the other entries are automatically determined by the symmetry condition.

The independent entries are:

1. The elements on the main diagonal: $A_{11}$ (represented by $a$), $A_{22}$ (represented by $d$), $A_{33}$ (represented by $f$). There are 3 such elements.
2. The elements in the upper triangular part (excluding the diagonal): $A_{12}$ (represented by $b$), $A_{13}$ (represented by $c$), $A_{23}$ (represented by $e$). There are 3 such elements.

Why these are independent:

• The diagonal elements ($A_{11}, A_{22}, A_{33}$) do not have a distinct symmetric counterpart ($A_{ii} = A_{ii}$), so their values can be chosen freely.
• For the off-diagonal elements, we only need to choose one from each pair $(A_{ij}, A_{ji})$. Conventionally, we choose the elements in the upper triangle (or lower triangle). For example, if we choose $A_{12}$, then $A_{21}$ is automatically set to be equal to $A_{12}$.

So, the total number of independent entries we need to fill is $3 + 3 = 6$. These are $a, b, c, d, e, f$.

General Tip: For an $n \times n$ symmetric matrix, the number of independent entries is given by the formula $\frac{n(n+1)}{2}$. For $n=3$, this is $\frac{3(3+1)}{2} = \frac{3 \times 4}{2} = 6$.

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Step 3: Counting Choices for Each Independent Entry

The problem states that all entries must be from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let's count the number of elements in this set. It contains 10 distinct integers.

Since each of the 6 independent entries ($a, b, c, d, e, f$) can take any value from this set, there are 10 choices for each of these 6 entries.

• Choices for $a$: 10 options
• Choices for $b$: 10 options
• Choices for $c$: 10 options
• Choices for $d$: 10 options
• Choices for $e$: 10 options
• Choices for $f$: 10 options

Why this step is taken: Each independent choice contributes to the total number of possible matrices. Since the choice for one entry does not affect the choices for other independent entries, we multiply the number of options for each.

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Step 4: Calculating the Total Number of Symmetric Matrices

To find the total number of distinct symmetric matrices, we use the Fundamental Principle of Counting (Multiplication Rule). Since there are 6 independent entries and each has 10 choices, the total number of ways to fill these entries is the product of the number of choices for each entry.

Total number of symmetric matrices = (Choices for $a$) $\times$ (Choices for $b$) $\times$ (Choices for $c$) $\times$ (Choices for $d$) $\times$ (Choices for $e$) $\times$ (Choices for $f$) Total number of symmetric matrices = $10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^6$.

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Important Tips and Common Mistakes to Avoid

• Understanding the Structure is Key: Always start by writing down the general matrix and applying the symmetry condition ($A_{ij} = A_{ji}$) to identify the independent elements. Don't just memorize a formula without understanding its derivation.
• Don't Overcount or Undercount: A common mistake is to assume all 9 elements of a $3 \times 3$ matrix are independent, leading to $10^9$. This is incorrect because $A_{12}$ and $A_{21}$ are not independent if the matrix is symmetric. Another mistake is to forget the diagonal elements, leading to $10^3$ or $10^6$ but for a $3 \times 3$ matrix with 9 elements, you'd only count 3 off-diagonal elements as independent. The correct count for independent elements is 6.
• Generalization: Remember the general formula for the number of $n \times n$ symmetric matrices with entries from a set of $k$ elements: $k^{n(n+1)/2}$. This can save time in similar problems.
• Distinguish from other matrix types: The number of matrices changes significantly for different types (e.g., skew-symmetric, diagonal, identity). Always refer back to the definition of the specific matrix type.

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Conclusion and Key Takeaway

By understanding the definition of a symmetric matrix ($A_{ij} = A_{ji}$), we determined that for a $3 \times 3$ matrix, there are only 6 independent entries that need to be chosen. Since each of these entries can take any of the 10 given values, the total number of possible symmetric matrices is $10^6$.

The final answer is $\boxed{10^{6}}$.