If A = \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix}, then A is
Step-by-Step Solution
Key Concept: Calculate A^2. If A^2 = 0, then A is a nilpotent matrix.
Given A = \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix}. <br> A^2 = \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix} \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix} <br> = \begin{pmatrix} (ab)(ab) + (b^2)(-a^2) & (ab)(b^2) + (b^2)(-ab) \\ (-a^2)(ab) + (-ab)(-a^2) & (-a^2)(b^2) + (-ab)(-ab) \end{pmatrix} <br> = \begin{pmatrix} a^2b^2 - a^2b^2 & ab^3 - ab^3 \\ -a^3b + a^3b & -a^2b^2 + a^2b^2 \end{pmatrix} <br> = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O. <br> Since A^2 = O, A is a nilpotent matrix.
Correct Answer: 3