Questions: Computing Determinants

Question 1

During row reduction of a matrix, you swap two rows to get a zero into a pivot position. How does this affect the determinant?

• A. It multiplies the determinant by 2, since you are rearranging two rows
• B. It negates the determinant — multiplies it by −1
• C. It leaves the determinant unchanged, since the same rows are still present
• D. It multiplies the determinant by the ratio of the two swapped rows

Answer: (B)

Swapping any two rows of a matrix negates its determinant (multiplies by −1). This is one of three fundamental row operation rules: row swap → negate det; multiply a row by scalar c → multiply det by c; add a multiple of one row to another → det unchanged. When computing det via row reduction, you must track every row swap and multiply the final result (product of the upper triangular diagonal) by (−1)^(number of swaps).

Question 2

A 3×3 matrix has three column vectors that all lie in the same plane through the origin. What is the determinant?

• A. 1, since the vectors form a valid basis for the plane
• B. 0, since the columns are linearly dependent and the matrix is singular
• C. The area of the triangle spanned by the three column tips
• D. Undefined, since three coplanar vectors cannot form a 3×3 matrix

Answer: (B)

The determinant measures the signed volume of the parallelepiped spanned by the column vectors. If all three columns lie in the same plane, the parallelepiped is completely flat — it has zero volume. This is also the algebraic signal: three coplanar vectors are linearly dependent (one is a linear combination of the others), and det(A) = 0 if and only if the matrix is singular. Zero determinant, singular matrix, and linearly dependent columns are all equivalent conditions.

Question 3

Adding a multiple of one row to another row during row reduction leaves the determinant unchanged.

• T. True
• F. False

Answer: True

This is one of three fundamental row operation rules, and the most useful: Gaussian elimination primarily uses this operation, so it preserves the determinant throughout. Only row swaps (which negate it) and row scaling (which multiply it by a scalar) change the value. You can row-reduce freely using add-a-multiple-of-a-row operations and only need to track swaps and scalings. This is why row reduction is efficient for computing determinants.

Question 4

If det(A) = 5 for a 3×3 matrix A, then the matrix 2A (most entry doubled) has determinant 10.

• T. True
• F. False

Answer: False

Multiplying every entry of an n×n matrix by scalar c multiplies the determinant by cⁿ, not c. Each row is scaled by c, and each row scaling multiplies the determinant by c. For a 3×3 matrix with c = 2: det(2A) = 2³ · det(A) = 8 · 5 = 40, not 10. This mistake comes from thinking of the determinant as linear in the matrix entries overall. It is multilinear — linear separately in each row — so scaling all n rows by c raises c to the nth power.

Question 5

Why does det(A) = 0 imply that the matrix A is singular (non-invertible)?

Model answer: The determinant measures the signed volume of the parallelepiped spanned by the column vectors of A. If det(A) = 0, the columns span a lower-dimensional space — they are linearly dependent, meaning at least one column is a linear combination of the others. A linear transformation with linearly dependent columns squashes space onto a lower-dimensional subspace: it is not one-to-one, so no inverse can exist. Algebraically: the system Ax = 0 has non-trivial solutions, the null space is non-trivial, and A cannot be inverted.

The equivalence runs in both directions: det(A) ≠ 0 iff A is invertible iff the columns are linearly independent iff Ax = b has a unique solution for every b. All of these conditions are equivalent, and the determinant provides a single scalar test for all of them. This is why the determinant appears throughout linear algebra — in Cramer's rule, eigenvalue computation, and the matrix inverse formula — as the fundamental indicator of whether a linear transformation preserves or collapses dimensionality.