Key Concepts and Formulas

This problem involves the properties of symmetric matrices, matrix multiplication, and solving a Diophantine equation (an equation where only integer solutions are sought).

1. Symmetric Matrix: A square matrix $A$ is symmetric if $A^T = A$, meaning its entries are symmetric with respect to the main diagonal. For a 2x2 matrix, this implies the off-diagonal elements are equal.
2. Matrix Multiplication: For two 2x2 matrices $P = \begin{pmatrix} p & q \\ r & s \end{pmatrix}$ and $Q = \begin{pmatrix} t & u \\ v & w \end{pmatrix}$, their product $PQ$ is given by: $PQ = \begin{pmatrix} pt+qv & pu+qw \\ rt+sv & ru+sw \end{pmatrix}$
3. Trace of a Matrix: The trace of a square matrix is the sum of its diagonal elements. For a matrix $M = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}$, $\text{Trace}(M) = m_{11} + m_{22}$.

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Step-by-Step Solution

1. Representing the Symmetric Matrix A

The problem states that A is a symmetric matrix of order 2 with integer entries. Let the matrix A be: $A = \begin{pmatrix} a & b \\ d & c \end{pmatrix}$ For A to be symmetric, its transpose $A^T$ must be equal to A. $A^T = \begin{pmatrix} a & d \\ b & c \end{pmatrix}$ Since $A = A^T$, we must have $b=d$. Therefore, a general symmetric matrix of order 2 can be written as: $A = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ Here, $a, b, c$ are integers, as specified in the problem statement. This is a crucial constraint for finding solutions.

2. Calculating $A^2$

Next, we need to find $A^2$, which is $A \times A$. $A^2 = \begin{pmatrix} a & b \\ b & c \end{pmatrix} \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ Using the rules of matrix multiplication: The element in the first row, first column of $A^2$ is $(a \times a) + (b \times b) = a^2 + b^2$. The element in the first row, second column of $A^2$ is $(a \times b) + (b \times c) = ab + bc$. The element in the second row, first column of $A^2$ is $(b \times a) + (c \times b) = ba + cb = ab + bc$. The element in the second row, second column of $A^2$ is $(b \times b) + (c \times c) = b^2 + c^2$. So, $A^2$ is: $A^2 = \begin{pmatrix} a^2 + b^2 & ab + bc \\ ab + bc & b^2 + c^2 \end{pmatrix}$

3. Finding the Sum of Diagonal Elements of $A^2$

The problem states that the sum of the diagonal elements of $A^2$ is 1. The diagonal elements of $A^2$ are $(a^2 + b^2)$ and $(b^2 + c^2)$. Sum of diagonal elements (Trace of $A^2$) $= (a^2 + b^2) + (b^2 + c^2)$. Equating this to 1: $a^2 + b^2 + b^2 + c^2 = 1$ $a^2 + 2b^2 + c^2 = 1$

4. Solving the Diophantine Equation

We need to find integer solutions for $a, b, c$ for the equation $a^2 + 2b^2 + c^2 = 1$. Since $a, b, c$ are integers, their squares $a^2, b^2, c^2$ must be non-negative integers ($0, 1, 4, 9, \dots$).

Let's analyze the possible values for $b$:

• Case 1: If $b \neq 0$ If $b$ is any non-zero integer, then $b^2 \ge 1$. This would mean $2b^2 \ge 2$. If $2b^2 \ge 2$, then $a^2 + 2b^2 + c^2 \ge 0 + 2 + 0 = 2$ (since $a^2 \ge 0$ and $c^2 \ge 0$). This contradicts our equation $a^2 + 2b^2 + c^2 = 1$. Therefore, $b$ cannot be a non-zero integer.

• Case 2: If $b = 0$ If $b=0$, then $b^2 = 0$. Substituting $b=0$ into the equation $a^2 + 2b^2 + c^2 = 1$: $a^2 + 2(0)^2 + c^2 = 1$ $a^2 + c^2 = 1$ Now we need to find integer solutions for $a$ and $c$ such that $a^2 + c^2 = 1$. Again, $a^2$ and $c^2$ must be non-negative integers.

  Let's consider possible values for $a^2$ and $c^2$:

  • If $a^2 = 1$, then $c^2 = 1 - 1 = 0$.

    • $a^2 = 1 \implies a = \pm 1$.
    • $c^2 = 0 \implies c = 0$. This gives two possible sets of values for $(a, c)$: $(1, 0)$ and $(-1, 0)$.

  • If $a^2 = 0$, then $c^2 = 1 - 0 = 1$.

    • $a^2 = 0 \implies a = 0$.
    • $c^2 = 1 \implies c = \pm 1$. This gives two possible sets of values for $(a, c)$: $(0, 1)$ and $(0, -1)$.

  • Are there other possibilities? If $a^2 \ge 2$ or $c^2 \ge 2$, then $a^2+c^2 \ge 2$, which would contradict $a^2+c^2=1$. So, these are all the possible integer solutions.

5. Listing the Possible Matrices A

Combining $b=0$ with the four pairs of $(a,c)$ solutions:

1. If $(a, c) = (1, 0)$ and $b=0$: $A_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
2. If $(a, c) = (-1, 0)$ and $b=0$: $A_2 = \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix}$
3. If $(a, c) = (0, 1)$ and $b=0$: $A_3 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$
4. If $(a, c) = (0, -1)$ and $b=0$: $A_4 = \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix}$

These are the four distinct matrices that satisfy all the given conditions.

6. Counting the Number of Matrices

We have found 4 distinct matrices.

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Tips and Common Mistakes to Avoid

• Integer Constraint: Always remember that $a, b, c$ must be integers. This is critical for solving $a^2 + 2b^2 + c^2 = 1$. If $a, b, c$ were real numbers, there would be infinitely many solutions.
• Symmetry: Do not forget the symmetric matrix condition ($b=d$). If you start with a general 2x2 matrix $\begin{pmatrix} a & b \\ d & c \end{pmatrix}$, you would get $a^2+bd+b^2+c^2=1$. But then you'd need to impose $b=d$ to simplify it to $a^2+2b^2+c^2=1$.
• Negative Solutions: When solving $x^2 = k$, remember that $x = \pm \sqrt{k}$. Many students forget the negative solutions, especially for $a^2=1$ and $c^2=1$.
• Systematic Approach for Diophantine Equations: For equations like $a^2 + 2b^2 + c^2 = 1$, start by analyzing the term with the largest coefficient (here, $2b^2$) or the term that limits possibilities most severely, as it helps narrow down the cases quickly.

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Summary and Key Takeaway

This problem effectively tests your understanding of matrix properties and your ability to solve simple Diophantine equations. By systematically defining the symmetric matrix, performing matrix multiplication, and then carefully analyzing the integer solutions for the resulting quadratic equation, we found exactly 4 possible matrices. The key steps involved recognizing the symmetric form, calculating the trace of $A^2$, and then exhaustively finding integer solutions for $a^2+2b^2+c^2=1$ by considering the constraints on $b$.

The final answer is $\boxed{\text{4}}$.