Exercise 3: Translating a Point With Homogeneous Coordinates — Possible Solution
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GIVEN
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Point: [2, -1]
Translation: (tx, ty) = (-3, 4)

STEP 1: THE 3x3 TRANSLATION MATRIX
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T = [[1, 0, -3],
     [0, 1,  4],
     [0, 0,  1]]

STEP 2: THE HOMOGENEOUS FORM OF THE POINT
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Padding [2, -1] with an extra 1 gives the homogeneous vector:
  [2, -1, 1]

STEP 3: THE FULL MATRIX-VECTOR MULTIPLICATION
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Row 0: (1)(2) + (0)(-1) + (-3)(1) = 2 + 0 - 3 = -1
Row 1: (0)(2) + (1)(-1) + (4)(1)  = 0 - 1 + 4  = 3
Row 2: (0)(2) + (0)(-1) + (1)(1)  = 0 + 0 + 1  = 1

T [2, -1, 1] = [-1, 3, 1]

Dropping the trailing 1 gives the translated point: (-1, 3), which
matches the direct arithmetic check (2 + (-3), -1 + 4) = (-1, 3).

WHY NO ORDINARY 2x2 MATRIX COULD DO THIS
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Per this chapter's own argument, any 2x2 matrix M always sends the
origin to itself: M x [0, 0] = [0, 0], because every entry of the
result is a sum of products where one factor is always 0. Translation
by (-3, 4) sends the origin to (-3, 4), not to itself - so no 2x2
matrix, no matter what numbers it contains, could ever reproduce that
behavior. The extra row and column in the 3x3 homogeneous matrix,
combined with the constant 1 appended to every vector, is specifically
what allows a nonzero shift to appear in the result even when the
starting point is the origin.

WHY THIS WORKS AS AN ANSWER
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It builds the translation matrix and homogeneous vector directly from
this chapter's own template, works through the full multiplication
entry by entry, cross-checks the result against simple direct
addition, and explains the necessity of the 3x3 homogeneous form using
this chapter's own origin-fixing argument rather than just asserting
that a 2x2 matrix "wouldn't work."