Question

If $A= \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix}$ and $A^{2018} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $(a + d)$ equals

• A. 1+i
• B. 0
• C. 2
• D. 2018

Answer 1

The Correct Option is C

Given matrix: $$ A = \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix} $$ We are asked to find the value of \(a + d\) where: $$ A^{2018} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ Step 1: Find the powers of matrix A. The matrix \( A \) is a 2x2 upper triangular matrix. We will explore the pattern of powers of \( A \) to find \( A^{2018} \). The first few powers of \( A \) are: - \( A^1 = A = \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix} \) - \( A^2 = \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix} \cdot \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} \) - \( A^3 = \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 0 & -i \end{bmatrix} \) - \( A^4 = \begin{bmatrix} 1 & 1 \\ 0 & i \end{bmatrix} \cdot \begin{bmatrix} 1 & 3 \\ 0 & -i \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 0 & 1 \end{bmatrix} \) Step 2: Observe the pattern of powers. From the above calculations, we observe the following pattern: - The top-left element of \( A^n \) is always 1. - The top-right element of \( A^n \) increases linearly by 1 each time, i.e., \( 1, 2, 3, 4, \dots \). - The bottom-left element is always 0. - The bottom-right element alternates between powers of \(i\), i.e., \( i, -1, -i, 1, \dots \). Therefore, the general form of \( A^n \) is: $$ A^n = \begin{bmatrix} 1 & n \\ 0 & i^n \end{bmatrix} $$ Step 3: Apply the pattern to \( A^{2018} \). From the pattern, we see that for even \(n\), \( i^n = 1 \). Since \( 2018 \) is even, we have: $$ A^{2018} = \begin{bmatrix} 1 & 2018 \\ 0 & 1 \end{bmatrix} $$ Step 4: Find \(a + d\). In the matrix \( A^{2018} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), we have \(a = 1\) and \(d = 1\). Therefore: $$ a + d = 1 + 1 = 2 $$ Conclusion: The value of \(a + d\) is \(2\). Thus, the correct answer is \( 2 \).