Exercise 2: A Rectangular Matrix Times a Column Vector — Possible Solution
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GIVEN
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J = [[1, 0, 2], [3, 1, 1]]   (a 2x3 matrix)
K = [[2], [1], [4]]           (a 3x1 matrix - a column vector)

STEP 1: CHECKING THE DIMENSION RULE FOR J x K
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J is 2x3 (2 rows, 3 columns). K is 3x1 (3 rows, 1 column). J's number
of columns (3) matches K's number of rows (3), so J x K is defined,
and per this chapter's own dimension rule the result will be a
2x1 matrix (J's row count x K's column count).

STEP 2: COMPUTING J x K
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Row 0: (1)(2) + (0)(1) + (2)(4) = 2 + 0 + 8  = 10
Row 1: (3)(2) + (1)(1) + (1)(4) = 6 + 1 + 4  = 11

J x K = [[10], [11]]  — a 2x1 matrix, as predicted.

STEP 3: WHY K x J IS NOT DEFINED
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For K x J, K's number of columns must match J's number of rows. K is
3x1, so its number of columns is 1. J is 2x3, so its number of rows
is 2. Since 1 does not equal 2, this chapter's own dimension rule is
violated - there's no way to take a "row of K, column of J" dot
product, because K's rows only have 1 entry each while J's columns
have 2 entries each; the dot product itself would be undefined term
by term. K x J is simply not a valid operation for these two shapes.

WHY THIS WORKS AS AN ANSWER
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It checks the dimension rule explicitly before computing anything,
carries out J x K entry by entry to confirm the predicted 2x1 shape,
and explains K x J's invalidity by pointing to the specific mismatched
numbers (K's 1 column vs. J's 2 rows) rather than simply asserting
that the order "doesn't work."