Find a nonzero 2 × 2 matrix A satisfying A2 = 0.

To find a nonzero \( 2 \times 2 \) matrix \( A \) such that \( A^2 = 0 \), we need a matrix whose square is the zero matrix, yet the matrix itself is not the zero matrix.

Let's consider a general \( 2 \times 2 \) matrix:
\[
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

We want to find \( A \) such that:
\[
A^2 = A \cdot A = 0
\]
This means the product of \( A \) with itself should yield the zero matrix:
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
\]

Carrying out the matrix multiplication:
\[
A^2 = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix}
\]
We require each element of this resulting matrix to be zero:
\[
\begin{aligned}
a^2 + bc &= 0 \\
ab + bd &= 0 \\
ac + cd &= 0 \\
bc + d^2 &= 0
\end{aligned}
\]

To satisfy these equations, we can consider some simple values. Let’s try:
\[
A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}
\]

Now, calculate \( A^2 \):
\[
A^2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + 1 \cdot 0 & 0 \cdot 1 + 1 \cdot 0 \\ 0 \cdot 0 + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
\]

Thus, \( A^2 = 0 \), and \( A \) is a nonzero matrix. Therefore, \( A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \) satisfies the condition \( A^2 = 0 \).

Would you like further details or have any questions?

Here are some related questions:
1. How would you find a matrix \( A \) such that \( A^2 = A \)?
2. Can you find a \( 3 \times 3 \) matrix that satisfies \( A^2 = 0 \)?
3. What are the eigenvalues of the matrix \( A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \)?
4. Can \( A^2 = 0 \) have non-zero determinant? Why or why not?
5. How would you generalize this to higher dimensions?

**Tip:** A matrix that squares to zero but is not the zero matrix is called a **nilpotent matrix**.

Find a nonzero 2 × 2 matrix A satisfying A^2 = 0.

Explore how to find a nonzero 2x2 matrix A such that A^2 = 0. This problem involves matrix multiplication and the concept of nilpotent matrices, with detailed steps and examples for advanced high school and undergraduate students.

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