This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — To show that A^4 = I_2, we first need to calculate A^2 and then A^4. Given the matrix A = i & 0 \\ 0 & -i . Recall that i^2 = -1. Step 1: Calculate A^2 A^2 = A × A A^2 = i & 0 \\ 0 & -i × i & 0 \\ 0 & -i Perform matrix multiplication: A^2 = (i)(i) + (0)(0) & (i)(0) + (0)(-i) \\ (0)(i) + (-i)(0) & (0)(0) + (-i)(-i) A^2 = i^2 + 0 & 0 + 0 \\ 0 + 0 & 0 + (-i)^2 Since i^2 = -1 and (-i)^2 = (-1)^2 i^2 = 1 · (-1) = -1: A^2 = -1 & 0 \\ 0 & -1 Step 2: Calculate A^4 We can calculate A^4 as A^2 × A^2. A^4 = -1 & 0 \\ 0 & -1 × -1 & 0 \\ 0 & -1 Perform matrix multiplication: A^4 = (-1)(-1) + (0)(0) & (-1)(0) + (0)(-1) \\ (0)(-1) + (-1)(0) & (0)(0) + (-1)(-1) A^4 = 1 + 0 & 0 + 0 \\ 0 + 0 & 0 + 1 A^4 = 1 & 0 \\ 0 & 1 Step 3: Compare with I_2 The identity matrix of order 2 is I_2 = 1 & 0 \\ 0 & 1 . Since A^4 = 1 & 0 \\ 0 & 1 , we have A^4 = I_2. Hence, it is proven that A^4 = I_2. What's next?