Key Concepts and Formulas

• Matrix Addition and Multiplication: Matrices of the same dimensions can be added by summing their corresponding elements. Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix. The element in the $i$-th row and $j$-th column of the product matrix is obtained by taking the dot product of the $i$-th row of the first matrix and the $j$-th column of the second matrix.
• Non-Commutativity of Matrix Multiplication: For matrices $A$ and $B$, in general, $AB \neq BA$. This is a crucial distinction from scalar algebra.
• Expansion of $(A+B)^2$ for Matrices: Due to non-commutativity, the expansion is $(A+B)^2 = (A+B)(A+B) = A(A+B) + B(A+B) = A^2 + AB + BA + B^2$. It is NOT $A^2 + 2AB + B^2$.
• Equality of Matrices: Two matrices are equal if and only if they have the same dimensions and all their corresponding elements are equal.

Step-by-Step Solution

Step 1: Understand the Problem and Identify Given Matrices

We are given two $2 \times 2$ matrices, $A$ and $B$, involving parameters $\alpha$ and $\beta$: $A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right] \quad \text{and} \quad B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right]$ The problem asks us to find two specific values of $\alpha$, denoted $\alpha_1$ and $\alpha_2$, based on two different matrix equations. Finally, we need to calculate the absolute difference $|\alpha_1 - \alpha_2|$.

The two conditions are:

1. $(\mathrm{A}+\mathrm{B})^{2}=\mathrm{A}^{2}+\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ (to find $\alpha_1$)
2. $(\mathrm{A}+\mathrm{B})^{2}=\mathrm{B}^{2}$ (to find $\alpha_2$)

Step 2: Calculate Essential Matrix Products

To simplify the subsequent calculations, we will first compute $A^2$, $B^2$, $AB$, and $BA$. This systematic approach prevents redundant calculations and reduces the chance of errors.

1. Calculate $A^2$: $A^2 = A \cdot A = \left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right] \left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right] = \left[\begin{array}{cc}(1)(1)+(-1)(2) & (1)(-1)+(-1)(\alpha) \\ (2)(1)+(\alpha)(2) & (2)(-1)+(\alpha)(\alpha)\end{array}\right]$ $A^2 = \left[\begin{array}{cc}1-2 & -1-\alpha \\ 2+2\alpha & -2+\alpha^2\end{array}\right] = \left[\begin{array}{cc}-1 & -1-\alpha \\ 2+2\alpha & \alpha^2-2\end{array}\right]$

2. Calculate $B^2$: $B^2 = B \cdot B = \left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right] \left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right] = \left[\begin{array}{cc}(\beta)(\beta)+(1)(1) & (\beta)(1)+(1)(0) \\ (1)(\beta)+(0)(1) & (1)(1)+(0)(0)\end{array}\right]$ $B^2 = \left[\begin{array}{cc}\beta^2+1 & \beta \\ \beta & 1\end{array}\right]$

3. Calculate $AB$: $AB = \left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right] \left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right] = \left[\begin{array}{cc}(1)(\beta)+(-1)(1) & (1)(1)+(-1)(0) \\ (2)(\beta)+(\alpha)(1) & (2)(1)+(\alpha)(0)\end{array}\right]$ $AB = \left[\begin{array}{cc}\beta-1 & 1 \\ 2\beta+\alpha & 2\end{array}\right]$

4. Calculate $BA$: $BA = \left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right] \left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right] = \left[\begin{array}{cc}(\beta)(1)+(1)(2) & (\beta)(-1)+(1)(\alpha) \\ (1)(1)+(0)(2) & (1)(-1)+(0)(\alpha)\end{array}\right]$ $BA = \left[\begin{array}{cc}\beta+2 & -\beta+\alpha \\ 1 & -1\end{array}\right]$

Step 3: Determine $\alpha_1$ using the first condition

The first condition for $\alpha_1$ is: $(A+B)^{2}=\mathrm{A}^{2}+\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ We substitute the correct expansion for $(A+B)^2$, which is $A^2 + AB + BA + B^2$. This is crucial because matrix multiplication is not commutative. $A^2 + AB + BA + B^2 = A^2 + \left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ Now, we can subtract $A^2$ from both sides of the equation, simplifying it: $AB + BA + B^2 = \left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ Next, we substitute the matrix expressions calculated in Step 2: $\left[\begin{array}{cc}\beta-1 & 1 \\ 2\beta+\alpha & 2\end{array}\right] + \left[\begin{array}{cc}\beta+2 & -\beta+\alpha \\ 1 & -1\end{array}\right] + \left[\begin{array}{cc}\beta^2+1 & \beta \\ \beta & 1\end{array}\right] = \left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ Perform the matrix addition by adding corresponding elements: $\left[\begin{array}{cc}(\beta-1)+(\beta+2)+(\beta^2+1) & 1+(-\beta+\alpha)+\beta \\ (2\beta+\alpha)+1+\beta & 2+(-1)+1\end{array}\right] = \left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ Simplify the left-hand side matrix: $\left[\begin{array}{cc}\beta^2+2\beta+2 & \alpha+1 \\ 3\beta+\alpha+1 & 2\end{array}\right] = \left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ For these two matrices to be equal, their corresponding elements must be equal. We can form a system of scalar equations:

1. Equating the $(1,2)$ elements: $\alpha+1 = 2 \implies \alpha = 1$.
2. Equating the $(1,1)$ elements: $\beta^2+2\beta+2 = 2 \implies \beta^2+2\beta = 0 \implies \beta(\beta+2) = 0$. This gives $\beta=0$ or $\beta=-2$.
3. Equating the $(2,1)$ elements: $3\beta+\alpha+1 = 2$. Substitute $\alpha=1$: $3\beta+1+1 = 2 \implies 3\beta+2 = 2 \implies 3\beta = 0 \implies \beta = 0$.
4. Equating the $(2,2)$ elements: $2 = 2$, which is consistent.

For the matrix equality to hold, $\beta$ must be $0$. This value is consistent with one of the possibilities from the $(1,1)$ element. Therefore, for the first condition, $\alpha_1 = 1$.

Step 4: Determine $\alpha_2$ using the second condition

The second condition for $\alpha_2$ is: $(A+B)^{2}=\mathrm{B}^{2}$ Again, we substitute the correct expansion for $(A+B)^2$: $A^2 + AB + BA + B^2 = B^2$ Subtract $B^2$ from both sides: $A^2 + AB + BA = O$ where $O$ is the $2 \times 2$ zero matrix $\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$. Now, substitute the matrix expressions calculated in Step 2: $\left[\begin{array}{cc}-1 & -1-\alpha \\ 2+2\alpha & \alpha^2-2\end{array}\right] + \left[\begin{array}{cc}\beta-1 & 1 \\ 2\beta+\alpha & 2\end{array}\right] + \left[\begin{array}{cc}\beta+2 & -\beta+\alpha \\ 1 & -1\end{array}\right] = \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$ Perform the matrix addition: $\left[\begin{array}{cc}(-1)+(\beta-1)+(\beta+2) & (-1-\alpha)+1+(-\beta+\alpha) \\ (2+2\alpha)+(2\beta+\alpha)+1 & (\alpha^2-2)+2+(-1)\end{array}\right] = \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$ Simplify the left-hand side matrix: $\left[\begin{array}{cc}2\beta & -\beta \\ 3\alpha+2\beta+3 & \alpha^2-1\end{array}\right] = \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$ Equate corresponding elements to zero:

1. Equating the $(1,1)$ elements: $2\beta = 0 \implies \beta = 0$.
2. Equating the $(1,2)$ elements: $-\beta = 0 \implies \beta = 0$. (Consistent)
3. Equating the $(2,2)$ elements: $\alpha^2-1 = 0 \implies \alpha^2 = 1 \implies \alpha = \pm 1$.
4. Equating the $(2,1)$ elements: $3\alpha+2\beta+3 = 0$. Substitute $\beta=0$: $3\alpha+2(0)+3 = 0 \implies 3\alpha+3 = 0 \implies 3\alpha = -3 \implies \alpha = -1$.

For the matrix equality to hold, $\alpha$ must be $-1$. This value is consistent with one of the possibilities from the $(2,2)$ element. Therefore, for the second condition, $\alpha_2 = -1$.

Step 5: Calculate $|\alpha_1 - \alpha_2|$

We have found $\alpha_1 = 1$ and $\alpha_2 = -1$. Now we compute the required absolute difference: $|\alpha_1 - \alpha_2| = |1 - (-1)| = |1+1| = |2| = 2$

Common Mistakes & Tips

• Incorrect Expansion of $(A+B)^2$: The most common mistake is to assume $(A+B)^2 = A^2 + 2AB + B^2$. Always remember that matrix multiplication is generally not commutative, leading to the correct expansion $A^2 + AB + BA + B^2$.
• Arithmetic Errors: Matrix calculations involve many additions and multiplications. Be meticulous with each step, especially when combining terms or dealing with negative signs.
• Inconsistent Values: If, when equating elements, you find conflicting values for $\alpha$ or $\beta$ (e.g., $\alpha=1$ from one element and $\alpha=2$ from another, assuming no calculation error), it indicates a logical inconsistency in the problem statement or your interpretation. Here, values were consistent.

Summary

This problem required a thorough understanding of matrix algebra, particularly the correct expansion of $(A+B)^2$ due to the non-commutative nature of matrix multiplication. We systematically calculated necessary matrix products ($A^2, B^2, AB, BA$) first. Then, we applied these to two given matrix equations, simplifying them and equating corresponding elements to solve for $\alpha$ and $\beta$. This process yielded $\alpha_1=1$ and $\alpha_2=-1$. Finally, we calculated the absolute difference $|\alpha_1 - \alpha_2|$, which was $2$.

The final answer is \boxed{2}