Linear Algebra! Vector Question! Urgent!

[molmath2000]

Anyone know where to start on this question? Any help would be very appreciated !! Linear Algebra and Vectors.
the question is on the image below!

 

[Steven G]

I would start at the beginning. What are the definitions of those indexes?

 

[HallsofIvy]

Note that matrices in N are of the form \(\displaystyle \begin{pmatrix}-3d & 2c \\ c & d\end{pmatrix}\) while matrices in M are of the form \(\displaystyle \begin{pmatrix}p & q \\ -q & r \end{pmatrix}\).

Any matrix in \(\displaystyle M\cap N\) must have -3d= p, 2c= q, c= -q, and d=r.

Now, what does "M:R" mean? Is it the dimension of M as a vector space over R?