Question:

Step 1: Compute $M^2$ by matrix multiplication. We calculate: $$M^2 = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$$ Multiplying the matrices: - Top-left: $0 \cdot 0 + i \cdot i = i^2 = -1$ - Top-right: $0 \cdot i + i \cdot 0 = 0$ - Bottom-left: $i \cdot 0 + 0 \cdot i = 0$ - Bottom-right: $i \cdot i + 0 \cdot 0 = i^2 = -1$ Therefore: $$M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -I$$ Step 2: Compute $M^3$ using the result from Step 1. Since $M^3 = M^2 \cdot M = (-I) \cdot M$: $$M^3 = -M = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}$$ Step 3: Compute $M^4$ to identify the periodicity. Since $M^4 = M^2 \cdot M^2 = (-I)(-I) = I$: $$M^4 = I$$ Step 4: Recognize the periodic pattern of powers of $M$. The powers of $M$ repeat with period 4: $$M^0 = I, \quad M^1 = M, \quad M^2 = -I, \quad M^3 = -M, \quad M^4 = I, \quad \ldots$$ Step 5: Calculate the sum of one complete period. One complete cycle of four consecutive powers sums to: $$M^0 + M^1 + M^2 + M^3 = I + M + (-I) + (-M) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$ Step 6: Determine how many complete periods fit in the sum from $M^0$ to $M^{2010}$. The total number of terms is $2010 - 0 + 1 = 2011$ terms. Dividing by the period: $2011 = 4 \times 502 + 3$ This means we have 502 complete periods (each summing to zero) plus 3 remaining terms. Step 7: Calculate the sum of the remaining terms. The remaining terms are $M^0, M^1, M^2$: $$S = 502 \times 0 + (M^0 + M^1 + M^2) = I + M + (-I) = M$$ Step 8: State the final answer. $$S = M = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$$ The answer is **Option 2**: $\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$