To solve the problem, we need to find the sum of all elements of the matrix \( A^2 \) given that \( A^3 = 0 \).
Let's denote the matrix \( A \) as follows:
\[
A = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}
\]
### Step 1: Understand the implication of \( A^3 = 0 \)
Since \( A^3 = 0 \), it means that multiplying the matrix \( A \) by itself three times results in the zero matrix. This can be expressed as:
\[
A^3 = A \cdot A \cdot A = 0
\]
### Step 2: Express \( A^3 \) in terms of \( A^2 \)
From the equation \( A^3 = A^2 \cdot A = 0 \), we can deduce that either \( A^2 = 0 \) or \( A \) is not invertible. However, since \( A^3 = 0 \), we will explore the implications of \( A^2 \).
### Step 3: Calculate \( A^2 \)
To find \( A^2 \), we compute the product \( A \cdot A \):
\[
A^2 = A \cdot A = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \cdot \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}
\]
Calculating the elements of \( A^2 \):
- The element at (1,1):
\[
\alpha^2 + \beta \gamma
\]
- The element at (1,2):
\[
\alpha \beta + \beta \delta
\]
- The element at (2,1):
\[
\gamma \alpha + \delta \gamma
\]
- The element at (2,2):
\[
\gamma \beta + \delta^2
\]
Thus, we have:
\[
A^2 = \begin{pmatrix} \alpha^2 + \beta \gamma & \alpha \beta + \beta \delta \\ \gamma \alpha + \delta \gamma & \gamma \beta + \delta^2 \end{pmatrix}
\]
### Step 4: Set \( A^2 \) to equal the zero matrix
Since \( A^3 = 0 \), we know that \( A^2 \cdot A = 0 \). This implies that each element of \( A^2 \) must satisfy certain conditions that lead to the zero matrix:
\[
A^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
\]
This gives us the following equations:
1. \( \alpha^2 + \beta \gamma = 0 \)
2. \( \alpha \beta + \beta \delta = 0 \)
3. \( \gamma \alpha + \delta \gamma = 0 \)
4. \( \gamma \beta + \delta^2 = 0 \)
### Step 5: Solve the equations
From these equations, we can conclude that:
- If \( \alpha, \beta, \gamma, \delta \) are all zero, then all equations hold true.
Thus, we can conclude:
\[
\alpha = 0, \beta = 0, \gamma = 0, \delta = 0
\]
### Step 6: Find the sum of all elements of \( A^2 \)
Since all elements of \( A^2 \) are zero:
\[
A^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
\]
The sum of all elements of \( A^2 \) is:
\[
0 + 0 + 0 + 0 = 0
\]
### Final Answer:
The sum of all the elements of \( A^2 \) is \( 0 \).
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