By inspection, find the inverse of the given one to one matrix operator 
By inspection, find the inverse of the given one to one matrix
  operator
1.The reflection about the $xy-plane$ in $R^3$
2.The dilation by a factor of 5 in $R^2$
3.The reflection about the $z-axis$ in $R^3$
4.The contraction by a factor of $\frac{1}{3}$ in $R^3$
5.The rotaion through an angle of $\frac{\pi}{3}$ in $R^3$

How to inspect it? Can someone just give some hint?
 A: Hints:
a) The transformations here are


*

*reflection

*dilation, contraction (scaling)

*rotation


These are linear transformations, so you can restrict yourself to $\mathbb{R}^{2\times 2}$ matrices for the 2D vectors and $\mathbb{R}^{3\times 3}$ for the 3D vectors. 
(The more general case would involve affine transformations, and thus homogenous coordinates needing an extra coordinate to use convenient matrices for the transformations)
I would as well interpret "inspection" as "try it out". So you could think how a matrix $A$ of the above kind would act on an input vector $x = (x_i)$ to produce the wanted output vector $y = A x$, with $y_i = \sum_j a_{ij} x_j$.
b) Let us look at the easy one, 2.: We need to find a matrix $A$ such that
$$
A x = 5 x \iff \\
\begin{pmatrix}
5 x_1 \\
5 x_2
\end{pmatrix}
= 
\begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
= 
\begin{pmatrix}
a_{11} x_1 & a_{12} x_2 \\
a_{21} x_1 & a_{22} x_2
\end{pmatrix}
$$
you could now figure out the coefficients $a_{ij}$ from component-wise comparison. Then you could derive the inverse from it.
Alternative: What is the inverse of increasing by a factor 5? Using the operation and then its inverse must give the original vector. You could go directly for the matrix of the inverse operation.
c) Now try 1.:
Here a vector has three coordinates $x = (x_1, x_2, x_3)^t$. A reflection at the $x$-$y$-plane affects the $z$-coordinate. So the problem is to find an $A$ such that
$$
A (x_1, x_2, x_3)^t = (x_1, x_2, -x_3)^t
$$
The plane itself consists of the vectors $(x_1, x_2, 0)^t$ which will not be changed by the reflection.
d) Regarding 3.: If we mirror a vector $x$ at some axis $a$, we get an image vector $y$ and expect the axis $a$ to bisect the line orthogonal which connects $x$ and $y$. A vector from the origin to a point on the axis should not be changed by the transformation.
e) Regarding 4.: If you solved 2. this should be a piece of cake. The additional dimension should be easy to handle.
f) Regarding 5.: @EmilioNovati pointed already out in the comments below, that this problem lacks information to nail it down to a unique solution. You need to know around which axis you rotate. 
Rotations themselves have the property that they preserve the length of the vectors. This they share with reflections. Also there is a restriction on how the coordinates are permuted, the determinant of the matrix must be $+1$. 
The easy case is to figure out how a 2D vector will get changed by a rotation in the $x$-$y$-plane (around the $z$-axis). This idea will extend to the 3D case and has to involve extra transformations if the axis of rotation is not parallel to one of the coordinate system axis or is not through the origin. 
A: I think "by inspection" in this case means "think about the geometry - no need to do any computing". For example:


*

*If you reflect and then reflect again over the same plane, you're back where you started, so reflection is its own inverse.

*If you expand everything by a factor of 5 then to get back to where you started you just ...
(can you finish now?)
