Exercise 1: Computing and Verifying an Inverse — Possible Solution
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GIVEN
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M = [[3, 2], [1, 4]]

STEP 1: THE DETERMINANT
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det(M) = (3)(4) - (2)(1) = 12 - 2 = 10

Since 10 is nonzero, M is not singular, and an inverse exists.

STEP 2: COMPUTING THE INVERSE
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Using A^-1 = (1/det(A)) x [[d, -b], [-c, a]] with a=3, b=2, c=1, d=4:

  Swap diagonal, negate off-diagonal: [[4, -2], [-1, 3]]
  Multiply by 1/10:

  M^-1 = [[4/10, -2/10], [-1/10, 3/10]]
       = [[0.4, -0.2], [-0.1, 0.3]]

STEP 3: VERIFYING M x M^-1 = I
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Row 0, Col 0: (3)(0.4) + (2)(-0.1)  = 1.2 - 0.2 = 1
Row 0, Col 1: (3)(-0.2) + (2)(0.3)  = -0.6 + 0.6 = 0
Row 1, Col 0: (1)(0.4) + (4)(-0.1)  = 0.4 - 0.4 = 0
Row 1, Col 1: (1)(-0.2) + (4)(0.3)  = -0.2 + 1.2 = 1

M x M^-1 = [[1, 0], [0, 1]] = I, confirming the inverse is correct.

WHY THIS WORKS AS AN ANSWER
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The determinant is checked first (per this chapter's own requirement
that it be nonzero before an inverse can exist), the inverse is built
using this chapter's own 2x2 formula with each step shown separately,
and the result is independently verified by computing the full matrix
product and confirming it equals the identity matrix, rather than
trusting the formula was applied correctly.