Matrices & Determinants
Nilpotent Matrices
Grade 12
Question:
<p>Let M be a \(3 \times 3\) matrix satisfying \(M^3 = 0\). Then which of the following statement(s) are true:</p>
<p>(a) \(\frac{1}{2}M^2 + M + I \neq 0\)</p>
<p>(b) \(\frac{1}{2}M^2 - M + I = 0\)</p>
<p>(c) \(\frac{1}{2}M^2 + M + I = 0\)</p>
<p>(d) \(\frac{1}{2}M^2 - M + I \neq 0\)</p>
Step-by-Step Solution
Key Concept: Use the nilpotent property $M^3 = 0$ to determine invertibility of related matrix expressions.
<p>Since $M^3 = 0$, we have $(I - M)(I + M + \frac{1}{2}M^2) = I - \frac{1}{2}M^3 = I$.</p><p>This means $I + M + \frac{1}{2}M^2$ is invertible, so $\frac{1}{2}M^2 + M + I \neq 0$, making (a) true.</p><p>Similarly, checking (b) and (c) shows they lead to contradictions with $M^3 = 0$. Statement (d) follows from (a).</p>
Correct Answer: a, d