Matrix A is a 3×3 matrix with det(A) = 5. What is det(3A)?
A15 — multiply det(A) by the scalar 3
B45 — multiply det(A) by 3², since there are two dimensions affected
C135 — multiply det(A) by 3³, since scaling a 3×3 matrix by 3 scales each of three dimensions
D5 — scalar multiplication of a matrix doesn't affect its determinant
det(cA) = cⁿ det(A) for an n×n matrix. For a 3×3 matrix, det(3A) = 3³ · det(A) = 27 · 5 = 135. The geometric explanation: the determinant measures volume. Scaling every entry by 3 scales each of the three spatial dimensions by 3, so volume scales by 3 × 3 × 3 = 27. The most common error is option A — treating det as a linear function of the scalar, which ignores the n-dimensional nature of the volume scaling.
Question 2 Multiple Choice
During Gaussian elimination on matrix A, you perform these operations in order: swap two rows, multiply one row by 5, add a multiple of one row to another. How does each operation affect det(A)?
ASwap negates det; scaling by 5 multiplies det by 5; row addition multiplies det by the row multiple used
BSwap negates det; scaling by 5 multiplies det by 5; row addition leaves det unchanged
CAll three operations are elementary row operations and therefore all leave det unchanged
DSwap does not affect det; scaling multiplies by 5; row addition leaves det unchanged
Row swap negates the determinant (reverses orientation of the parallelogram/parallelepiped). Scaling a row by c multiplies det by c (stretches one edge, scaling the volume proportionally). Adding a multiple of one row to another is a shear — it distorts the shape but preserves area/volume, so det is unchanged. This is precisely why Gaussian elimination computes determinants efficiently: track only the row swaps (sign changes) and scalings, and multiply the final triangular diagonal.
Question 3 True / False
det(Aᵀ) = det(A) for any square matrix A, meaning a matrix and its transpose always have the same determinant.
TTrue
FFalse
Answer: True
This is a fundamental property. It reflects a deep symmetry: rows and columns contribute equally to the determinant's geometric meaning. One consequence is that every column property of determinants has an equivalent row property — for example, det is zero if any row is a zero row OR if any column is a zero column, and linear dependence in rows implies the same as linear dependence in columns.
Question 4 True / False
If det(A) = 0, it means matrix A has no eigenvalues, which is why A is not invertible.
TTrue
FFalse
Answer: False
This reverses the relationship. det(A) = 0 means 0 IS an eigenvalue of A — there exists a nonzero vector v such that Av = 0·v = 0, meaning A maps v to the zero vector. This nontrivial null space is exactly what makes A non-invertible. The correct statement is: det(A) = 0 implies 0 is an eigenvalue, not that A lacks eigenvalues. (A matrix always has eigenvalues over the complex numbers, and most non-invertible matrices have many nonzero eigenvalues in addition to zero.)
Question 5 Short Answer
Why does adding a multiple of one row to another row leave the determinant unchanged, even though it alters the entries of the matrix?
Think about your answer, then reveal below.
Model answer: Adding a multiple of one row to another is geometrically a shear transformation. Shearing a parallelogram (or parallelepiped) distorts its shape — edges tilt — but preserves its area (or volume), because the base length and perpendicular height remain the same. Since the determinant measures signed volume, and shear preserves volume, the determinant is unchanged. This is analogous to how you can shear a rectangle into a parallelogram without changing its area as long as the base and height stay constant.
Understanding this geometrically rather than algebraically is what makes the rule intuitive rather than arbitrary. It also explains why Gaussian elimination (which consists of row operations, including shears) can compute the determinant without changing its magnitude — you only need to track the sign changes from row swaps and the multiplicative factor from row scaling.