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?
The characteristic polynomial of M is $$\lambda^2-2\lambda+1=(\lambda-1)^2.$$ Hence M has the single eigenvalue 1 with algebraic multiplicity 2 and is similar to the Jordan block $$\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$ Therefore statement (A) is true. Since M=I+N where $$N=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}$$ and N²=0, we have $$M^k=I+kN.$$ Thus $$M^{26}=I+26N=\begin{bmatrix}27&-26\\26&-25\end{bmatrix},$$ so p=27, q=-26, r=26 and s=-25. The determinant is ps-qr=1, therefore the matrix is unimodular and for every pair of integers (m,n), there exists a unique integer solution (x,y). Hence (C) is true. Also, $$\sum_{k=1}^{26}M^k=26I+\left(\sum_{k=1}^{26}k\right)N=26I+351N,$$ giving $$\begin{bmatrix}377&-351\\351&-325\end{bmatrix}.$$ Therefore a=377, not 378, so (B) is false. Let A denote this matrix. Then $$A+tI=\begin{bmatrix}377+t&-351\\351&-325+t\end{bmatrix}.$$ Its determinant is $$\det(A+tI)=(377+t)(-325+t)+351^2=t^2+52t=t(t+52).$$ For every positive real t, this determinant is nonzero, so the system has a unique solution. Hence (D) is true. Therefore the correct options are (A), (C) and (D).
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