I need some help with aspects of Example 1.4

The relevant text from Gibson's book is as follows:

**Question 1**In the above text, Gibson writes the following:

" ... ... Then a brief calculation verifies that any point \(\displaystyle p + t(q - p) = (1- t)p + tq\) also lies on the line ... ... "

I am unable to perform the brief calculation that Gibson refers to ... can someone please help me by showing the calculation and how it works ... ...

**Question 2**

" ... Since at least one of a,b is non-zero, the system has a non-trivial solution, By linear algebra the \(\displaystyle 3 x 3\) matrix of coefficients is singular, so the rows \(\displaystyle (p_1, p_2 , 1) , ( q_1, q_2, 1) , ( r_1, r_2, 1)\) are linearly independent. ... ... "

Can someone lease explain how we know that the \(\displaystyle 3 \times 3\) matrix of coefficients is singular?

Hope someone can help with the above two questions ...

Peter