Partition of a Matrix In Linear Algebra, we have been taught that the partition of a matrix $A$ consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to be equal to the original matrix. But their elements are different, as one contains scalars and the other matrices. Please help me understand.

You have to concatenate the matrices. Here's an example:
$$A= \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, B = \begin{pmatrix} 10 \\ 20 \end{pmatrix}, C = (100,200), D = 1000,$$
then
$$\begin{pmatrix}A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} & \begin{pmatrix} 10 \\ 20 \end{pmatrix} \\ \begin{pmatrix} 100 & 200\end{pmatrix} & 1000 \end{pmatrix} \equiv  \begin{pmatrix} 1 & 2 & 10 \\ 3 & 4 & 20 \\ 100 & 2000 & 1000 \end{pmatrix}.$$
