Key Concept: Matrix Multiplication

The fundamental operation required to solve this problem is matrix multiplication. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. If we have two $2 \times 2$ matrices, say $P = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix}$ and $Q = \begin{bmatrix} q_{11} & q_{12} \\ q_{21} & q_{22} \end{bmatrix}$, their product $PQ$ is given by: $PQ = \begin{bmatrix} (p_{11}q_{11} + p_{12}q_{21}) & (p_{11}q_{12} + p_{12}q_{22}) \\ (p_{21}q_{11} + p_{22}q_{21}) & (p_{21}q_{12} + p_{22}q_{22}) \end{bmatrix}$ Each element in the product matrix is obtained by taking the dot product of a row from the first matrix and a column from the second matrix. Specifically, the element in the $i$-th row and $j$-th column of the product matrix is found by multiplying the $i$-th row of the first matrix by the $j$-th column of the second matrix, element by element, and summing the results.

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Problem Setup

We are given a matrix $A$ as: $A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}$ We are also given that $A^2$ is of the form: $A^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix}$ Our goal is to calculate $A^2$ by multiplying $A$ by itself, and then compare the resulting matrix with the given form of $A^2$ to determine the values of $\alpha$ and $\beta$.

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Step-by-Step Calculation of $A^2$

To find $A^2$, we need to multiply matrix $A$ by itself: $A^2 = A \cdot A = \begin{bmatrix} a & b \\ b & a \end{bmatrix} \begin{bmatrix} a & b \\ b & a \end{bmatrix}$

Let's calculate each element of the product matrix step-by-step:

1. Element in the first row, first column ($C_{11}$):

   • Why: To find the element in the first row, first column of the product matrix, we take the dot product of the first row of the first matrix ($A$) and the first column of the second matrix ($A$).
   • Calculation: $C_{11} = (a)(a) + (b)(b) = a^2 + b^2$

2. Element in the first row, second column ($C_{12}$):

   • Why: To find the element in the first row, second column, we take the dot product of the first row of the first matrix ($A$) and the second column of the second matrix ($A$).
   • Calculation: $C_{12} = (a)(b) + (b)(a) = ab + ba = 2ab$

3. Element in the second row, first column ($C_{21}$):

   • Why: To find the element in the second row, first column, we take the dot product of the second row of the first matrix ($A$) and the first column of the second matrix ($A$).
   • Calculation: $C_{21} = (b)(a) + (a)(b) = ba + ab = 2ab$

4. Element in the second row, second column ($C_{22}$):

   • Why: To find the element in the second row, second column, we take the dot product of the second row of the first matrix ($A$) and the second column of the second matrix ($A$).
   • Calculation: $C_{22} = (b)(b) + (a)(a) = b^2 + a^2 = a^2 + b^2$

Now, we assemble these elements to form the matrix $A^2$: $A^2 = \begin{bmatrix} a^2 + b^2 & 2ab \\ 2ab & a^2 + b^2 \end{bmatrix}$

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Equating Elements and Finding $\alpha, \beta$

We are given that $A^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix}$. By comparing the elements of our calculated $A^2$ with the given form, we can directly find $\alpha$ and $\beta$:

Comparing the element in the first row, first column: $\alpha = a^2 + b^2$

Comparing the element in the first row, second column: $\beta = 2ab$

(Note: The elements in the second row, first column and second row, second column also consistently yield $\beta = 2ab$ and $\alpha = a^2+b^2$ respectively, confirming our results.)

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Conclusion and Option Selection

From our calculations, we have found that $\alpha = a^2 + b^2$ and $\beta = 2ab$. Matching these values with the given options: (A) $\alpha = 2ab,\,\beta = {a^2} + {b^2}$ (B) $\alpha = {a^2} + {b^2},\,\beta = ab$ (C) $\alpha = {a^2} + {b^2},\,\beta = 2ab$ (D) $\alpha = {a^2} + {b^2},\,\beta = {a^2} - {b^2}$

Our derived values match option (C).

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

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Tips for JEE Aspirants

1. Master Matrix Multiplication: This is a fundamental operation in matrices. Practice it until it becomes second nature. A common mistake is to multiply element-wise (like regular numbers), which is incorrect for matrix multiplication. Always remember "row by column".
2. Order Matters: Matrix multiplication is generally not commutative, meaning $AB \neq BA$. In this problem, we are squaring a matrix ($A \cdot A$), so the order is fixed.
3. Recognize Patterns: Notice that the given matrix $A$ is a symmetric matrix (elements $a_{ij} = a_{ji}$). The result $A^2$ is also symmetric. This can sometimes serve as a quick check for your calculations.
4. Careful with Algebra: Ensure you correctly sum and multiply the algebraic terms (e.g., $ab+ba=2ab$).
5. Time Management: For "easy" difficulty problems like this, ensure accuracy and speed. Don't rush, but be efficient with your steps.

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Summary and Key Takeaway

This problem effectively tests your understanding of basic matrix multiplication. By systematically applying the row-by-column rule for matrix multiplication, we calculated $A^2$ and then compared its elements with the given form to find the values of $\alpha$ and $\beta$. The key takeaway is the precise application of the matrix multiplication definition to correctly compute the elements of the product matrix.