Riemann surface
matlab

$1 c=m a$ and $d=m b$ lead to $a d=a m b=b c$. With no zeros, $a d=b c$ is the equation for a $2 \times 2$ matrix to have rank 1 .

The equations $1c = ma$ and $d = mb$ can be rearranged to obtain $ad = am(b) = bc$. Without any zeros, the equation $ad = bc$ represents a condition for a $2 \times 2$ matrix to have rank 1.

In this context, let’s consider a $2 \times 2$ matrix:

$$\begin{bmatrix} a & b \ c & d \ \end{bmatrix}$$

If the matrix has rank 1, it means that the rows or columns are linearly dependent, and one of them can be expressed as a scalar multiple of the other. Let’s assume that the first row is a scalar multiple of the second row, so we have:

$$\begin{bmatrix} a & b \ c & d \ \end{bmatrix} = m \cdot \begin{bmatrix} c & d \ c’ & d’ \ \end{bmatrix}$$

where $m$ is a scalar and $c’, d’$ are non-zero elements. Expanding this equation gives:

\begin{aligned} a &= mc \ b &= md \ \end{aligned}

Comparing these equations with the given equations $1c = ma$ and $d = mb$, we see that they are equivalent. Therefore, the condition $ad = bc$ is satisfied for a $2 \times 2$ matrix to have rank 1.

2 The three edges going around the triangle are $\boldsymbol{u}=(5,0), \boldsymbol{v}=(-5,12), \boldsymbol{w}=(0,-12)$. Their sum is $u+v+w=(0,0)$. Their lengths are $|\boldsymbol{u}|=5,|\boldsymbol{v}|=13,|\boldsymbol{w}|=12$. This is a $5-12-13$ right triangle with $5^2+12^2=25+144=169=13^2$ – the best numbers after the $3-4-5$ right triangle.

The three edges going around the triangle are $\boldsymbol{u} = (5, 0)$, $\boldsymbol{v} = (-5, 12)$, and $\boldsymbol{w} = (0, -12)$. Their sum is given by $u + v + w = (0, 0)$. The lengths of these edges can be calculated as follows: $|\boldsymbol{u}| = 5$, $|\boldsymbol{v}| = 13$, and $|\boldsymbol{w}| = 12$.

It is observed that these edge lengths form a $5-12-13$ right triangle. This can be verified by checking that $5^2 + 12^2 = 25 + 144 = 169 = 13^2$, which satisfies the Pythagorean theorem. The $5-12-13$ right triangle is a well-known right triangle with particularly nice number properties, second only to the famous $3-4-5$ right triangle.

Hence, based on the given edge lengths, the triangle formed by $\boldsymbol{u}$, $\boldsymbol{v}$, and $\boldsymbol{w}$ is a $5-12-13$ right triangle.

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