a)

Determinants.

Here is the formula for the determinant of a $2 \times 2$ matrix:
\[M = \begin{pmatrix} a & b \\ c&d \end{pmatrix} \Rightarrow \mathrm{det}\left(M\right) = ad-bc \]

$\mathrm{det}\left(A\right) = \simplify[]{{a11}*{a22}-{a12}*{a21} = {dA}}$
$\mathrm{det}\left(B\right) = \simplify[]{{b11}*{b22}-{b12}*{b21} = {dB}}$
$\mathrm{det}\left(C\right) = \simplify[]{{c11}*{c22}-{c12}*{c21} = {dC}}$

Determinant of a product of matrices.

If we have two $n \times n$ matrices $M$ and $N$ then:
\[\mathrm{det}\left(MN\right) = \mathrm{det}\left(M\right)\mathrm{det}\left(N\right)\]
And it follows that if we have a third matrix $P$ that:
\[\mathrm{det}\left(MNP\right) = \mathrm{det}\left(M\right)\mathrm{det}\left(N\right)\mathrm{det}\left(P\right)\]

Thus for our example we have:

\[\begin{eqnarray*}\mathrm{det}\left(ABC\right) &=& \mathrm{det}\left(A\right)\times\mathrm{det}\left(B\right)\times\mathrm{det}\left(C\right)\\ &=& \var{dA}\times \var{dB} \times \var{dC}\\ &=& \var{dA*dB*dC} \end{eqnarray*} \]

Inverse of a $2 \times 2$ matrix

Suppose $M$ is a $2 \times 2$ matrix and $\mathrm{det}\left(M\right) = \Delta \neq 0$.

Then $M$ is invertible and:
\[M = \begin{pmatrix} a & b \\ c&d \end{pmatrix} \Rightarrow M^{-1} = \frac{1}{\Delta} \begin{pmatrix} d & -b\\ -c& a \end{pmatrix}=\begin{pmatrix} \frac{d}{\Delta} & -\frac{b}{\Delta}\\ -\frac{c}{\Delta}& \frac{a}{\Delta} \end{pmatrix}\]

Applying this to these examples we obtain:

b)

\[A^{-1} = \begin{pmatrix} \simplify[std]{{a22}/{dA}} &\simplify[std]{{-a12}/{dA}}\\\simplify[std]{{-a21}/{dA}}&\simplify[std]{{a11}/{dA}}\end{pmatrix}\]

c)

\[B^{-1} = \begin{pmatrix} \simplify[std]{{b22}/{dB}} &\simplify[std]{{-b12}/{dB}}\\\simplify[std]{{-b21}/{dB}}&\simplify[std]{{b11}/{dB}}\end{pmatrix}\]

d)

\[C^{-1} = \begin{pmatrix} \simplify[std]{{c22}/{dC}} &\simplify[std]{{-c12}/{dC}}\\\simplify[std]{{-c21}/{dC}}&\simplify[std]{{c11}/{dC}}\end{pmatrix}\]