Matrices & Determinants
Types of Matrices
Grade 12
Question:
<p>If \(\mathbf{A} = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}\), then <strong>A</strong> is a/an</p>
<p>(a) idempotent matrix</p>
<p>(b) nilpotent matrix</p>
<p>(c) involutory matrix</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: A nilpotent matrix of order n satisfies A^n = O. Calculate A² and verify it equals the zero matrix.
<p><strong>Solution:</strong></p><p>Given: $\mathbf{A} = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}$</p><p>Calculate $\mathbf{A}^2$:</p><p>$\mathbf{A}^2 = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix} \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}$</p><p>$= \begin{bmatrix} a^2b^2 - a^2b^2 & ab^3 - ab^3 \\ -a^3b + a^3b & -a^2b^2 + a^2b^2 \end{bmatrix}$</p><p>$= \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \mathbf{O}$</p><p>Since $\mathbf{A}^2 = \mathbf{O}$, <strong>A</strong> is a nilpotent matrix of order 2.</p>
Correct Answer: b