Here is a detailed, step-by-step solution to the problem, incorporating explanations, tips, and an alternative method using the Cayley-Hamilton Theorem.

1. Understanding the Problem and Key Concepts

The problem asks us to find the sum $\alpha + \beta$ given a matrix $A$ and a matrix equation $\alpha A^2 + \beta A = 2I$. To solve this, we will primarily use the following matrix operations:

• Matrix Multiplication: To calculate $A^2$.
• Scalar Multiplication of Matrices: To find $\alpha A^2$ and $\beta A$.
• Matrix Addition: To sum $\alpha A^2$ and $\beta A$.
• Matrix Equality: To equate the resulting matrix with $2I$ and form a system of linear equations.
• Identity Matrix ($I$): For a $2 \times 2$ matrix, $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Thus, $2I = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$.

An alternative, often more elegant approach for problems involving powers of a matrix, is the Cayley-Hamilton Theorem, which states that every square matrix satisfies its own characteristic equation