Exercise 3: Recognizing Singularity From the Columns Themselves — Possible Solution
====================================================================

GIVEN
------------------------------
N = [[6, 3], [4, 2]]

STEP 1: THE DETERMINANT
------------------------------
det(N) = (6)(2) - (3)(4) = 12 - 12 = 0

Since the determinant is exactly 0, N is singular and has no inverse,
per this chapter's own singularity rule.

STEP 2: LOOKING AT THE TWO COLUMNS
------------------------------
N's first column is [6, 4]. N's second column is [3, 2].

Checking whether the second column is a scalar multiple of the first:
  6 x 0.5 = 3   (matches the second column's first entry)
  4 x 0.5 = 2   (matches the second column's second entry)

Both entries match exactly with a scalar of 0.5, so column 2 = 0.5 x
column 1 - the two columns point along the exact same direction (one
is just a shorter version of the other).

WHY THIS ALWAYS PRODUCES A SINGULAR MATRIX
------------------------------
Per this chapter's own geometric picture, a matrix transforms the
unit square into a parallelogram whose sides are given by the
matrix's own columns. If one column is a scalar multiple of the
other, the two "sides" of that parallelogram point in the same
direction - there's no genuine second dimension to the shape, so it
has been flattened into a single line rather than a real
parallelogram with area. A flattened parallelogram has zero area,
which is exactly what a zero determinant reports. This means the
singularity could have been predicted just by noticing the columns
were parallel, without doing the determinant arithmetic at all -
the arithmetic simply confirms what the column relationship already
guaranteed.

WHY THIS WORKS AS AN ANSWER
------------------------------
It computes the determinant using this chapter's own formula to
confirm singularity, then independently verifies the same conclusion
by finding the exact scalar relationship between the two columns, and
connects that relationship back to this chapter's own area-of-a-
parallelogram geometric picture to explain why parallel columns and a
zero determinant are really the same fact seen two different ways.