If M = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>5</mn><mo>/</mo><mn>2</mn></mtd><mtd><mn>3</mn><mo>/</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn><mo>/</mo><mn>2</mn></mtd><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math>, then which of the following matrices is equal to M<sup>2022</sup>?
(A) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3034</mn></mtd><mtd><mn>3033</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3033</mn></mtd><mtd><mo>-</mo><mn>3032</mn></mtd></mtr></mtable></mfenced></math>
(B) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3034</mn></mtd><mtd><mo>-</mo><mn>3033</mn></mtd></mtr><mtr><mtd><mn>3033</mn></mtd><mtd><mo>-</mo><mn>3032</mn></mtd></mtr></mtable></mfenced></math>
(C) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3033</mn></mtd><mtd><mn>3032</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3032</mn></mtd><mtd><mo>-</mo><mn>3031</mn></mtd></mtr></mtable></mfenced></math>
(D) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3032</mn></mtd><mtd><mn>3031</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3031</mn></mtd><mtd><mo>-</mo><mn>3030</mn></mtd></mtr></mtable></mfenced></math>
Step-by-Step Solution
Key Concept: Express M as I + N where N is a nilpotent matrix (N^2 = 0) and use the binomial expansion (I + N)^n = I + nN.
M = [[5/2, 3/2], [-3/2, 1/2]] = [[1, 0], [0, 1]] + [[3/2, 3/2], [-3/2, -3/2]]. Let I = [[1, 0], [0, 1]] and N = [[3/2, 3/2], [-3/2, -3/2]]. Note that N^2 = [[3/2, 3/2], [-3/2, -3/2]] * [[3/2, 3/2], [-3/2, -3/2]] = [[9/4-9/4, 9/4-9/4], [-9/4+9/4, -9/4+9/4]] = [[0, 0], [0, 0]]. Thus, M^n = (I + N)^n = I + nN = [[1, 0], [0, 1]] + n * [[3/2, 3/2], [-3/2, -3/2]] = [[1 + 3n/2, 3n/2], [-3n/2, 1 - 3n/2]]. For n = 2022, M^2022 = [[1 + 3(1011), 3(1011)], [-3(1011), 1 - 3(1011)]] = [[1 + 3033, 3033], [-3033, 1 - 3033]] = [[3034, 3033], [-3033, -3032]].
Correct Answer: 2