There exists a $2 \times 2$ invertible matrix $N$ with real entries such that

$$ MN = N \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $$

The value of $a$ is $378$

For any two given integers $m$ and $n$, there exist unique integers $x$ and $y$ such that

$$ px + qy = m \quad \text{and} \quad rx + sy = n $$

For each positive real number $t$, the system of linear equations

\begin{align*} (a + t)x + by &= 1 \\ cx + (d + t)y &= -1 \end{align*}

has a unique solution

The determinant of $Q - 2I$ is zero

The determinant of $Q - 6I$ is 12

The determinant of $Q - 3I$ is 15

$yz = 2$

For any given $(a, b, c) \in S$, the system of linear equations

$$ \begin{aligned} & a x+b y=1 \\ & b x+c y=-1 \end{aligned} $$

has a unique solution.

For any given $(a, b, c) \in S$, the system of linear equations

$$ \begin{aligned} & (a+1) x+b y=0 \\ & b x+(c+1) y=0 \end{aligned} $$

has a unique solution.