Matrices & Determinants
Characteristic Polynomial
Grade 12
Question:
<p>If <strong>A</strong> is a square matrix of order 3 and <strong>I</strong> is an Identity matrix of order 3 such that <strong>A</strong><sup>3</sup> − 2<strong>A</strong><sup>2</sup> − <strong>A</strong> + 2<strong>I</strong> = 0, then <strong>A</strong> is equal to</p>
<p>(a) <strong>I</strong></p>
<p>(b) 2<strong>I</strong></p>
<p>(c) \[\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{pmatrix}\]</p>
<p>(d) \[\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{pmatrix}\]</p>
Step-by-Step Solution
Key Concept: Verify which matrices satisfy the given matrix equation by substituting each option into A³ − 2A² − A + 2I = 0.
<p><strong>Step 1:</strong> Given characteristic equation: <strong>A</strong><sup>3</sup> − 2<strong>A</strong><sup>2</sup> − <strong>A</strong> + 2<strong>I</strong> = 0</p><p><strong>Step 2:</strong> Check option (a): If <strong>A</strong> = <strong>I</strong>, then <strong>I</strong> − 2<strong>I</strong> − <strong>I</strong> + 2<strong>I</strong> = 0 ✓</p><p><strong>Step 3:</strong> Check option (b): If <strong>A</strong> = 2<strong>I</strong>, then 8<strong>I</strong> − 2(4<strong>I</strong>) − 2<strong>I</strong> + 2<strong>I</strong> = 8<strong>I</strong> − 8<strong>I</strong> − 2<strong>I</strong> + 2<strong>I</strong> = 0 ✓</p><p><strong>Step 4:</strong> Check option (c): The characteristic equation becomes $λ^3 − 2λ^2 + λ − 2 = 0$, which is different from the given equation ✗</p><p><strong>Step 5:</strong> Check option (d): The characteristic equation becomes $λ^3 − 2λ^2 − λ + 2 = 0$, matching the given equation ✓</p><p>∴ Options (a), (b), and (d) satisfy the given matrix equation.</p>
Correct Answer: a,b,d