This explanation addresses the solution to the equation $A^3 = I_2$, where $A$ is a $2 \times 2$ matrix, and $I_2$ is the identity matrix of size 2. Here's a breakdown of the steps and reasoning:

1. Definition of $A^3 = I_2$:
   This means that multiplying $A$ by itself three times results in the identity matrix. $A$ must satisfy certain conditions for this to hold.

2. Inverse Matrix Calculation:
   The inverse of $A$, $A^{-1}$, is derived using the determinant formula: $A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.$ Using this, $A^2$ and eventually $A^3$ are computed.

3. Five Equations:
   From the matrix equality $A^3 = I_2$, we obtain five equations involving the entries of $A$:

   &ad - bc = 1, \quad (1) \\ &a^2 + bc = d, \quad (2) \\ &ab + bd = -b \implies b(a + d + 1) = 0, \quad (3) \\ &ca + dc = -c \implies c(a + d + 1) = 0, \quad (4) \\ &cb + d^2 = a. \quad (5) \end{aligned}$$

4. Case Analysis:
   To solve these equations, two cases arise based on $a + d + 1$:

   • Case 1: $a + d + 1 \neq 0$: From equations (3) and (4), $b = 0$ and $c = 0$. Substituting into (1) and (2), we find $a = d = 1$, resulting in $A = I_2$.
   • Case 2: $a + d + 1 = 0$: Here, $d = -a - 1$. Substituting into equations, one condition remains: $a^2 + a + 1 = -bc$. The matrix $A$ can then be expressed as: $A = \begin{bmatrix} a & b \\ -\frac{a^2 + a + 1}{b} & -1 - a \end{bmatrix}, \quad b \neq 0.$

5. Conclusion:
   This construction shows there are infinitely many solutions to $A^3 = I_2$, parameterized by real numbers $a$ and $b$ (with $b \neq 0$).

Would you like me to delve deeper into any step?

5 Questions to Expand:

1. Can I explain how the inverse formula for $A$ is derived?
2. Would you like a detailed breakdown of the determinant condition $ad - bc = 1$?
3. Should I simplify the derivation of the case $a + d + 1 \neq 0$?
4. Do you want to explore why $A$ results in infinitely many solutions?
5. Should I illustrate an example of such a matrix $A$?

Tip: For matrix equations, always verify the determinant is non-zero when working with inverses!