Singular matrix squared question

Question is:

A 2 x 2 singular matrix M is given as

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


Find M^2 and give your answer as a multiple of M.

OK, so I know that a singular matrix means determinant = 0, ie in this case ad = bc. Not sure how that helps. anyway, I do the matrix multiplication:

$\begin{pmatrix}a^2 + bc & ac + cd\\ab + bd& bc + d^2\end{pmatrix}$


The answer is M^2 = (a + d)M

How do they work that out?



The next part of question then asks:

Hence find a formula which gives M^n in terms of M.

I can't even do first part but answer is: M^n = (a + d)^n-1 M

I want to work out first part first - that might give me a clue as to this next step.

Well for part (i) you can just replace bc with ad!

Part (ii) can be done by induction on n - assume true for n = k and then multiply by M and use part (i)