Question.  How do I compute with matrices in MatrixForm?  After evaluating an expression such as

A = {{1, 2}, {3, 4}} // MatrixForm

I cannot get Mathematica to do any algebra with A.  For example, if I try

Det[A]

then Mathematica just spits back Det[A] without evaluating the determinant.

Answer.  Whereas a matrix is just a list of lists, the MatrixForm of a matrix is not a list of lists—not a matrix at all.  Rather, the result of an expression such as

MatrixForm[{{1, 2}, {3, 4}}]

or

{{1, 2}, {3, 4}} // MatrixForm

is merely a special display form (look at the special label of the output cell).  And in

A = {{1, 2}, {3, 4}} // MatrixForm

the MatrixForm function is applied before a value is assigned to A; in other words, this expression is the same as:

A = MatrixForm[{{1, 2}, {3, 4}}]

You need to keep the matrix as a matrix.  Here are two ways to do so and still see the matrix displayed as a matrix:

Method 1.

A = {{1, 2}, {3, 4}};
A // MatrixForm

Method 2.

MatrixForm[A = {{1, 2}, {3, 4}}]