Find two different 2 × 2 matrices A such that A

2 = 0 but A ̸= 0.

To find two different \( 2 \times 2 \) matrices \( A \) such that \( A^2 = 0 \) but \( A \neq 0 \), we need matrices where squaring them results in the zero matrix, but they themselves are not zero matrices.

### Example 1:
Consider the matrix
\[
A_1 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}
\]

Let's compute \( A_1^2 \):

\[
A_1^2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + 1 \cdot 0 & 0 \cdot 1 + 1 \cdot 0 \\ 0 \cdot 0 + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
\]

Thus, \( A_1^2 = 0 \), but \( A_1 \neq 0 \).

### Example 2:
Consider the matrix
\[
A_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}
\]

The computation for this matrix will be similar to the first example:
\[
A_2^2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 

Find two different 2 × 2 matrices A such that A^2 = 0 but A ≠ 0.

Explore two examples of 2x2 matrices A where A^2 = 0 but A is not the zero matrix. This problem demonstrates the concept of nilpotent matrices, a key topic in linear algebra. Suitable for undergraduate students studying linear algebra.

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