Compute det of N = [ [4, -2, 1], [0, 3, -1], [2, 5, 0] ].

• A. 36
• B. 18
• C. -18
• D. 0

Answer: (B)

Determinants tell you how a linear transformation scales area or volume, and for a 3×3 matrix you can compute it by expanding along a row into 2×2 determinants (cofactor expansion).

Compute along the first row: 4 times the determinant of the minor [[3, -1], [5, 0]] minus (-2) times the determinant of the minor [[0, -1], [2, 0]] plus 1 times the determinant of the minor [[0, 3], [2, 5]].

The minors are:

• det[[3, -1], [5, 0]] = 3·0 − (-1)·5 = 5

• det[[0, -1], [2, 0]] = 0·0 − (-1)·2 = 2

• det[[0, 3], [2, 5]] = 0·5 − 3·2 = -6

So the determinant is 4·5 − (-2)·2 + 1·(-6) = 20 + 4 − 6 = 18.

Since the determinant is nonzero, the transformation is invertible and preserves a nonzero volume scaling of 18.