How do I logically/conceptually approach this question in regards to Matrices, their Identities, and Partitions

Question

I have linked to the question I am not looking for a direct answer, rather I need help getting started. We are focusing on Matrices and their uses/properties. I think I have a good idea of Matrix Identities, but the wording kind of throws me off with this question. I have dyscalculia and that may be throwing me off with the wording. I kind of want a conceptual step by step explanation of what I need to do with this problem so I can grasp its meaning/importance. I want to be able to figure it out on my own. Thanks so much in advance!

Answer 1

For part $(a)$, you just need to apply the definition of matrix multiplication.

For part $(b)$, you need to know that if you have a big matrix $A$ and you partition it into four smaller matrices

$$ A = \begin{pmatrix} B && C \\ D && E \end{pmatrix} $$ then you can compute $A^2$ by multiplying the matrices just like you would have computed the power of a $2 \times 2$ matrix. That is,

$$ A^2 = \begin{pmatrix} B && C \\ D && E \end{pmatrix} \begin{pmatrix} B && C \\ D && E \end{pmatrix} = \begin{pmatrix} B^2 + CD & BC + CE \\ DB + ED & DC + E^2 \end{pmatrix} $$

whenever the multiplications of the matrices involved make sense. This works much more generally (you can find more details in the wiki page ) but that's all you need to know.