Question

For the matrices \( A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} \), if \( (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} \), then among the following which one is true?}

• A. \( x = 16, y = 3 \)
• B. \( x = 18, y = 11 \)
• C. \( x = 5, y = 7 \)
• D. \( x = 11, y = 2 \)

Hint:

For a \( 2 \times 2 \) matrix with characteristic equation \( (\lambda-1)^2 = 0 \), the powers follow an arithmetic progression in terms of \( n \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To solve this, we first need to find a general expression for \( A^n \). We can do this by looking at patterns or using the characteristic equation.

Step 2: Detailed Explanation:
Let's compute the first few powers of \( A \):
\[ A^1 = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \]
\[ A^2 = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 5 & -8 \\ 2 & -3 \end{bmatrix} \]
\[ A^3 = \begin{bmatrix} 5 & -8 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 7 & -12 \\ 3 & -5 \end{bmatrix} \]
By observation, the pattern for \( A^n \) is:
\[ A^n = \begin{bmatrix} 2n+1 & -4n \\ n & -2n+1 \end{bmatrix} \]
Substituting \( n = 15 \):
\[ A^{15} = \begin{bmatrix} 2(15)+1 & -4(15) \\ 15 & -2(15)+1 \end{bmatrix} = \begin{bmatrix} 31 & -60 \\ 15 & -29 \end{bmatrix} \]
Now, find \( A^{15} + B \):
\[ A^{15} + B = \begin{bmatrix} 31 & -60 \\ 15 & -29 \end{bmatrix} + \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} = \begin{bmatrix} 2 & -11
2 & -11 \end{bmatrix} \]
The given equation is:
\[ \begin{bmatrix} 2 & -11 \\ 2 & -11 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
This leads to the system of equations: \( 2x - 11y = 0 \).
Testing the options:
(D) If \( x = 11, y = 2 \), then \( 2(11) - 11(2) = 22 - 22 = 0 \).

Step 3: Final Answer:
The correct pair is \( x = 11, y = 2 \).