Question

Let \[ A = \begin{bmatrix} 2 & -3 \\ 1 & -2 \end{bmatrix}, \quad B = \begin{bmatrix} 14 & 21 \\ 7 & 10 \end{bmatrix}. \] If \[ (A^{4} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \] find (x, y).

• A. (0, 0)
• B. (1, -1)
• C. (2, -2)
• D. (-1, 1)

Hint:

If the determinant of the coefficient matrix of a homogeneous system is non-zero, the system has only the trivial solution.

Answer 1

The Correct Option is A

Step 1: Compute powers of matrix \(A\).
First, calculate \(A^2\): \[ A^2 = \begin{bmatrix} 2 & -3 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} 2 & -3 \\ 1 & -2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I. \]
Step 2: Find \(A^4\).
\[ A^4 = (A^2)^2 = I^2 = I. \]
Step 3: Evaluate \(A^4 + B\).
\[ A^4 + B = I + B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 14 & 21 \\ 7 & 10 \end{bmatrix} = \begin{bmatrix} 15 & 21 \\ 7 & 11 \end{bmatrix}. \]
Step 4: Solve the homogeneous system.
\[ \begin{bmatrix} 15 & 21 \\ 7 & 11 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \] The determinant is: \[ (15)(11) - (21)(7) = 165 - 147 = 18 \neq 0. \] Since the determinant is non-zero, the only solution is the trivial solution.
Step 5: Final conclusion.
\[ x = 0, \quad y = 0. \]