Matrices & Determinants
Idempotent matrix
Grade Class 12
Question:
<p>Consider the following statements</p><p><strong>Statement-1 :</strong> If <em>A</em> is an idempotent non-zero matrix and <em>I</em> is an identity matrix of the same order, such that (<em>A</em> + <em>I</em>)<sup><em>n</em></sup> = <em>I</em> + 127 <em>A</em>. (<em>n</em> ∈ N), then '<em>n</em>' has 3 positive divisors.</p><p><strong>Statement-2 :</strong> Let <em>A</em> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>6</mn><mi>x</mi></mtd></mtr></mtable></mfenced></math>, <em>B</em> = [<em>a</em> <em>b</em> <em>c</em>] and <em>C</em> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mtd><mtd><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mtd><mtd><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mtd><mtd><mn>2</mn><mi>x</mi></mtd><mtd><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>2</mn><mi>x</mi></mtd><mtd><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup></mtd><mtd><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced></math> be three given matrices, where <em>a</em>, <em>b</em>, <em>c</em> and <em>x</em> ∈ R. Given that tr(<em>AB</em>) = tr(<em>C</em>) ∀ <em>x</em> ∈ R, where tr(<em>A</em>) denotes trace of <em>A</em>. Solving, we get <em>a</em> + <em>b</em> + <em>c</em> = 7.</p><p>Then, which of the following options is/are correct ?</p>
(A) Statement-1 is True
(B) Statement-1 is False
(C) Statement-2 is True
(D) Statement-2 is False
Step-by-Step Solution
Key Concept: For Statement-1, use (A+I)^n = I + (2^n - 1)A for idempotent A. For Statement-2, calculate tr(AB) and tr(C) and equate coefficients of powers of x.
<p><strong>Statement-1:</strong> Since <em>A</em> is idempotent, <em>A</em><sup>2</sup> = <em>A</em>. Then (<em>A</em> + <em>I</em>)<sup><em>n</em></sup> = <em>I</em> + (2<sup><em>n</em></sup> - 1)<em>A</em>. Given 2<sup><em>n</em></sup> - 1 = 127, so 2<sup><em>n</em></sup> = 128 = 2<sup>7</sup>, hence <em>n</em> = 7. 7 is a prime number, so it has 2 divisors (1 and 7). Thus, Statement-1 is False.</p><p><strong>Statement-2:</strong> tr(<em>AB</em>) = tr(<em>BA</em>) = <em>a</em>(3<em>x</em><sup>2</sup>) + <em>b</em>(1) + <em>c</em>(6<em>x</em>). tr(<em>C</em>) = (<em>x</em>+2)<sup>2</sup> + 2<em>x</em> + 5<em>x</em><sup>2</sup> = <em>x</em><sup>2</sup> + 4<em>x</em> + 4 + 2<em>x</em> + 5<em>x</em><sup>2</sup> = 6<em>x</em><sup>2</sup> + 6<em>x</em> + 4. Equating coefficients: 3<em>a</em> = 6 \Rightarrow <em>a</em> = 2; 6<em>c</em> = 6 \Rightarrow <em>c</em> = 1; <em>b</em> = 4. Thus <em>a</em> + <em>b</em> + <em>c</em> = 2 + 4 + 1 = 7. Statement-2 is True.</p>
Correct Answer: 1, 3