Questions: Determinants of 2×2 and 3×3 Matrices

Question 1

A 2×2 matrix A transforms the unit square into a parallelogram with area 5. You then compute det(A) = −5. What does the negative sign indicate?

• A. The determinant was computed incorrectly — area cannot be negative
• B. The matrix is not invertible because the determinant is negative
• C. The transformation reversed orientation (like a reflection) while scaling area by a factor of 5
• D. The transformation shrinks area rather than expanding it

Answer: (C)

The absolute value |det(A)| = 5 gives the area scaling factor — the unit square maps to a parallelogram of area 5. The sign indicates orientation: positive means the transformation preserved handedness (rotation, scaling); negative means it reversed orientation (like a reflection or a shear that flips the axes). A determinant of −5 means area scales by 5 AND orientation flips. Neither sign makes the matrix non-invertible — det ≠ 0 in both cases means the matrix is invertible.

Question 2

You compute the determinant of a 3×3 matrix and get 0. What does this tell you about the geometric effect of the transformation?

• A. The transformation rotates 3D space by exactly 90 degrees
• B. The transformation preserves all distances but changes angles
• C. The transformation collapses 3D space into a lower-dimensional subspace, making it non-invertible
• D. The transformation scales all volumes by 0 but remains one-to-one

Answer: (C)

det(A) = 0 means the transformation maps the unit cube to a shape with zero volume — 3D space is collapsed into a plane, a line, or a point. Different input vectors get mapped to the same output (the transformation is not one-to-one), so it cannot be reversed. Option D contradicts itself: a transformation that collapses volume to zero cannot be one-to-one. This connection between det = 0 and non-invertibility is the fundamental application of determinants throughout linear algebra.

Question 3

For a 2×2 matrix, swapping its two rows produces a new matrix whose determinant is the negative of the original.

• T. True
• F. False

Answer: True

Row swapping reverses orientation, which changes the sign of the determinant. If the original rows form a parallelogram with a certain signed area, swapping the rows is equivalent to reflecting that parallelogram, reversing its orientation. More generally, any single row interchange multiplies the determinant by −1. This is one of the key properties that uniquely characterizes the determinant function: multilinearity in each row, antisymmetry under row interchange, and det(I) = 1.

Question 4

The Rule of Sarrus is a shortcut for computing determinants that generalizes to matrices larger than 3×3.

• T. True
• F. False

Answer: False

The Rule of Sarrus works only for 3×3 matrices. Applying its diagonal shortcut to 4×4 or larger matrices gives the wrong answer. For larger matrices, you must use cofactor expansion (Laplace expansion) or row reduction. This is a common source of error: students memorize Sarrus for 3×3 and incorrectly extend the diagonal trick to larger matrices.

Question 5

Explain the geometric meaning of the determinant of a 2×2 matrix, and why det(A) = 0 implies the matrix is not invertible.

Model answer: The determinant of a 2×2 matrix equals the signed area of the parallelogram formed by its two column vectors. It measures how the matrix scales area: a unit square transforms to a parallelogram of area |det(A)|. If det(A) = 0, the two column vectors are parallel (or one is zero), so the parallelogram collapses to a line with zero area. Geometrically, this means the transformation squashes 2D space into 1D — different input vectors map to the same output, so the transformation cannot be reversed.

The invertibility connection follows directly from the geometric picture: an invertible transformation must be one-to-one so it can be reversed. If the transformation collapses area to zero, it maps an entire line of input vectors to a single point — multiple inputs share the same output, which is irrecoverable. The formula det(A) = ad − bc captures this algebraically: if the columns are proportional, the anti-diagonal term bc equals the diagonal term ad, and their difference is zero.