Question

Let the matrix $M=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfy $M^n=M^{n-2}+M^2-I$ for $n=3,4,5,6,\dots$ where $I$ is the identity matrix of order $3$.
Also $U_1,U_2,U_3$ are column matrices satisfying $M^{50}{U}_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],M^{50}{U}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],M^{50}{U}_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ and $U$ is a $3\times 3$ matrix whose columns are $U_1,U_2,U_3$.
Then

• A. $M^{50}=25M^2-24I$
• B. $det\left(adj{M}^{50}\right)$ is equal to $1$
• C. Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 2\\ 4\end{array}\right]$ be two matrices. Then the system of equations $UX=B$ has infinitely many solutions.
• D. Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 2\\ 4\end{array}\right]$ be two matrices. Then the system of equations $UX=B$ has a unique solution.

Answer 1

Answer: (A), (B), (D)

Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 2\\ 4\end{array}\right]$ be two matrices. Then the system of equations $UX=B$ has a unique solution.

$M^{50}=M^{48}+M^2-I$
Further, $M^{48}=M^{46}+M^2-I$
$M^{48}=M^{46}+M^2-I$
$⋮$
$M^6=M^4+M^2-I$
$M^4=M^2+M^2-I$
Adding the above equations, we get
$M^{50}=25M^2-24I$

Here, $M^2=MM=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]$
$\Rightarrow M^{50}=\left[\begin{array}{ccc}25& 0& 0\\ 25& 25& 0\\ 25& 0& 25\end{array}\right]-24\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow M^{50}=\left[\begin{array}{ccc}1& 0& 0\\ 25& 1& 0\\ 25& 0& 1\end{array}\right]$
$\Rightarrow det\left(M^{50}\right)=1$
$\therefore det\left(adj{M}^{50}\right)=(det(M^{50}){)}^{3-1}=1$

Here, matrix $U=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow det\left(U\right)=1\ne 0$
$\Rightarrow$ The system of equations $UX=B$ has a unique solution.