If A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math>, then A<sup>2</sup> - 5A - 2I is equal to
Step-by-Step Solution
Key Concept: Use the Cayley-Hamilton theorem or direct matrix multiplication to compute A^2 and substitute into the expression.
Given A = [[1, 2], [3, 4]]. A^2 = [[1, 2], [3, 4]] * [[1, 2], [3, 4]] = [[1+6, 2+8], [3+12, 6+16]] = [[7, 10], [15, 22]]. 5A = [[5, 10], [15, 20]]. 2I = [[2, 0], [0, 2]]. A^2 - 5A - 2I = [[7-5-2, 10-10-0], [15-15-0, 22-20-2]] = [[0, 0], [0, 0]] = O.
Correct Answer: 1