Question

Consider the matrix \[ M= \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \] Let \(p,q,r,s,a,b,c,d\) be integers such that \[ M^{26}= \begin{bmatrix} p & q \\ r & s \end{bmatrix} \] and \[ \sum_{k=1}^{26} M^k= \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \] Then which of the following statements is (are) TRUE?

• A. There exists a \(2\times2\) invertible matrix \(N\) with real entries such that \[ MN= N \begin{bmatrix} 1 & 1 0 & 1 \end{bmatrix} \]
• B. The value of \(a\) is \(378\)
• C. For any two given integers \(m\) and \(n\), there exist unique integers \(x\) and \(y\) such that \[ px+qy=m \] and \[ rx+sy=n \]
• D. For each positive real number \(t\), the system of linear equations \[ (a+t)x+by=1 \] \[ cx+(d+t)y=-1 \] has a unique solution

Hint:

If: \[ A^2=0 \] then: \[ (I+A)^n=I+nA \] using binomial expansion.

Answer 1

The Correct Option is A