Consider the matrix $M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$ Let $p$, $q$, $r$, $s$, $a$, $b$, $c$ and $d$ 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?

1. A There exists a $2 \times 2$ invertible matrix $N$ with real entries such that $MN = N\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$
2. B The value of $a$ is $378$
3. C For any two given integers $m$ and $n$, there exist unique integers $x$ and $y$ such that $px + qy = m$ and $rx + sy = n$
4. 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

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Answers: A. There exists a $2 \times 2$ invertible matrix $N$ with real entries such that $MN = N\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ · 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

Determinants Consider the matrix $M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$ Let $p$, $q$, $r$, $s$, $a$, $b$, $c$ and $d$ 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? Watch this solved on the board The full walkthrough, written out a line at a time with narration explaining each step as it appears. Pause anywhere and ask why a line is there. Free to watch

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