Practice: The determinant

Self-check

Six short questions. Answer each one in your head (or on paper) before opening the collapsible. Trying to retrieve the answer is where the learning sticks; rereading feels productive but does much less.

1. What does the determinant of a matrix measure?

Show answer

The factor by which the transformation scales area (in 2D) or volume (in 3D). Every region, whatever its shape, scales by the same factor, because a linear transformation stretches space uniformly. After applying M, a region of area A has area det(M) · A.

2. What is the formula for the determinant of a 2x2 matrix [[a, b], [c, d]]?

Show answer

det = ad - bc: the main-diagonal product minus the off-diagonal product. This is the signed area of the parallelogram spanned by the two columns [a, c] and [b, d].

3. The determinant is “signed.” What two pieces of information does that single number carry?

Show answer

The magnitude is the size scaling (how much bigger or smaller areas get), and the sign tells you orientation: positive means orientation was preserved, negative means space was flipped over like a mirror reflection.

4. What does det = 0 mean geometrically, and what does it tell you about the columns?

Show answer

The transformation collapses the whole plane onto a line or a single point; all area is lost. It happens exactly when the two columns are linearly dependent (one is a scalar multiple of the other), so the parallelogram they span has zero area. A zero determinant is the signature of a collapse, and the transformation is not invertible.

5. What does a negative determinant tell you?

Show answer

That orientation was flipped (a reflection, space turned over). The magnitude still gives the area scaling, so det = -1 is a reflection that preserves area, not a shrink. The negative sign is a separate flag from the size.

6. What is the product rule for determinants, and why is it true without algebra?

Show answer

det(AB) = det(A) · det(B). If B scales areas by one factor and A by another, doing B then A scales by the product of the two. Scaling factors compose by multiplying, so the determinant of a composition is the product of the determinants.

Try it yourself, part 1: compute and interpret

For each matrix [[a, b], [c, d]], compute det = ad - bc, then say in a phrase what it means geometrically (areas grow, shrink, are preserved, collapse, or orientation flips). About 7 minutes, pen and paper.

a) [[4, 0], [0, 3]]
b) [[1, 2], [3, 4]]
c) [[2, 1], [4, 2]]
d) [[0, -1], [1, 0]]
e) [[3, 1], [2, 4]]
f) [[1, 0], [0, -1]]

Check your work

a) det = (4)(3) - (0)(0) = 12. Areas grow by a factor of 12 (a pure stretch).
b) det = (1)(4) - (2)(3) = 4 - 6 = -2. Magnitude 2 (areas double), negative sign (orientation flipped).
c) det = (2)(2) - (1)(4) = 4 - 4 = 0. Collapse: the columns [2, 4] and [1, 2] are dependent ([2,4] = 2·[1,2]), so space is crushed onto a line. Not invertible.
d) det = (0)(0) - (-1)(1) = 1. Area preserved (a rotation).
e) det = (3)(4) - (1)(2) = 12 - 2 = 10. Areas grow by a factor of 10.
f) det = (1)(-1) - (0)(0) = -1. Area preserved, orientation flipped (a reflection across the x-axis).

Try it yourself, part 2: the product rule and invertibility

About 8 minutes, pen and paper.

Step 1. Let A = [[2, 0], [0, 3]] and B = [[1, 1], [0, 1]]. Compute det(A) and det(B), then use the product rule to predict det(AB). Now compute the matrix AB (apply A to each column of B) and its determinant directly, and check that it matches.

Step 2. Which of these transformations can be run backward (are invertible)? [[2, 0], [0, 3]], [[2, 4], [1, 2]], [[1, 2], [3, 4]].

Check your work

Step 1. det(A) = (2)(3) - 0 = 6. det(B) = (1)(1) - (1)(0) = 1. Product rule predicts det(AB) = 6 · 1 = 6. Now compute AB: the columns of B are [1, 0] and [1, 1], and A · [x, y] = [2x, 3y], so A · [1, 0] = [2, 0] and A · [1, 1] = [2, 3]. Thus AB = [[2, 2], [0, 3]], and det(AB) = (2)(3) - (2)(0) = 6. It matches the product rule.

Step 2. A transformation is invertible exactly when its determinant is nonzero.

[[2, 0], [0, 3]]: det = 6 ≠ 0, invertible.
[[2, 4], [1, 2]]: det = 4 - 4 = 0, not invertible (it collapsed space onto a line).
[[1, 2], [3, 4]]: det = 4 - 6 = -2 ≠ 0, invertible (the negative sign just means it also flips orientation).

Flashcards

Ten cards. Click any card to reveal the answer. Use the Print flashcards button to lay out the full set as one card per page, ready to print or save as a PDF for offline review.

Q. What does the determinant of a matrix measure?

The factor by which the transformation scales area (2D) or volume (3D). The same factor applies to every region, because a linear transformation scales space uniformly: new area = det(M) times old area.

Q. What is the formula for a 2x2 determinant?

For [[a, b], [c, d]], det = ad - bc: main-diagonal product minus off-diagonal product. It is the signed area of the parallelogram spanned by the columns [a, c] and [b, d].

Q. Why is the determinant called 'signed', and what two things does it encode?

The magnitude is the size scaling (how much areas grow or shrink); the sign is orientation (positive = preserved, negative = flipped like a mirror). One number, two pieces of information.

Q. What does det = 0 mean, and what does it say about the columns?

The transformation collapses the plane onto a line or point (all area lost), which happens exactly when the columns are linearly dependent. A zero determinant means the transformation is not invertible.

Q. What does a negative determinant tell you?

Orientation was flipped (a reflection; space turned over). The magnitude still gives the area scaling, so det = -1 preserves area while flipping orientation, not a shrink.

Q. How do you read the determinant off the unit square?

The unit square (area 1) becomes the parallelogram spanned by the two columns. Since it started at area 1, the area of that parallelogram is the determinant. Plot the columns, measure the parallelogram.

Q. What is the determinant of a rotation? Of a shear?

Both are 1: area is preserved. A rotation spins space without resizing it; a shear distorts the square into a leaning parallelogram of the same base and height, so the area is unchanged even though the shape is not.

Q. What is the product rule for determinants?

det(AB) = det(A) · det(B). Scaling factors compose by multiplying: if B doubles areas and A triples them, doing B then A scales areas by six.

Q. What does the determinant mean in 3D?

The factor by which a 3x3 transformation scales volume, read as the volume of the parallelepiped the unit cube becomes. det = 0 means the cube was flattened (volume lost); negative means space was turned inside out.

Q. How does the determinant connect to invertibility?

det ≠ 0 means invertible: space was rearranged without collapsing, so the transformation can be run backward. det = 0 means a dimension was collapsed and cannot be recovered, so it is not invertible.