Matrices & Determinants
Matrix multiplication
Grade Class 12
Question:
If A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math>, then A<sup>20</sup> + (A<sup>T</sup>)<sup>20</sup> equals
A<sup>20</sup> + (A<sup>T</sup>)<sup>20</sup> = 2I + 20(A-I) + 190(A-I)<sup>2</sup>
A<sup>20</sup> + (A<sup>T</sup>)<sup>20</sup> = 2I + 20(A-I) + 200(A-I)<sup>2</sup>
A<sup>20</sup> + (A<sup>T</sup>)<sup>20</sup> = 2I + 20(A-I) + 210(A-I)<sup>2</sup>
A<sup>20</sup> + (A<sup>T</sup>)<sup>20</sup> = 2I + 20(A-I) + 220(A-I)<sup>2</sup>
Step-by-Step Solution
Key Concept: Decompose A into I + N where N is a nilpotent matrix, then use binomial expansion for A^n.
Let A = I + N, where N = [0 0 0; 2 0 0; 3 2 0]. Note that N^2 = [0 0 0; 0 0 0; 4 0 0] and N^3 = 0. Using binomial expansion, A^n = (I+N)^n = I + nN + n(n-1)/2 * N^2. Calculate A^20 and (A^T)^20 and sum them to find the result.
Correct Answer: 4