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Solution

For (A),

$\left[[2,0,0],[0,2,0],[0,0,1]\right]$

This is a scaling matrix.

Here, $S_x=2$ and $S_y=2$.

So, (A) matches with (II).

For (B),

$\left[[1,0,2],[0,1,2],[0,0,1]\right]$

This is a translation matrix.

Here, $T_x=2$ and $T_y=2$.

So, (B) matches with (IV).

For (C),

$\left[[0.5,-0.866,0],[0.866,0.5,0],[0,0,1]\right]$

The standard rotation matrix is:

$\left[[\cos\theta,-\sin\theta,0],[\sin\theta,\cos\theta,0],[0,0,1]\right]$

Here,

$\cos 60^\circ=0.5$

and

$\sin 60^\circ=0.866$

So, this matrix represents rotation by $60^\circ$.

Therefore, (C) matches with (III).

For (D),

$\left[[1,2,0],[0,1,0],[0,0,1]\right]$

This is shear in $x$ direction.

Here, shear factor is $2$.

So, (D) matches with (I).

Final matching is:

A-II, B-IV, C-III, D-I