Rotate then reflect, and the flag lands in one place. Reflect then rotate, and it lands somewhere else. Both compositions have determinant −1, so the determinant cannot tell you which one you did.
The same two matrices in the two possible orders give two different transformations and one determinant.
| Transformation | Matrix | Determinant |
|---|---|---|
| Reflection in the x-axis | (1, 0; 0, −1) | −1 |
| Reflection in the y-axis | (−1, 0; 0, 1) | −1 |
| Reflection in y = x | (0, 1; 1, 0) | −1 |
| Reflection in y = −x | (0, −1; −1, 0) | −1 |
| Reflection in y = x tanθ | (cos2θ, sin2θ; sin2θ, −cos2θ) | −1 |
| Rotation through θ anticlockwise | (cosθ, −sinθ; sinθ, cosθ) | 1 |
| Stretch, factor p horizontally, q vertically | (p, 0; 0, q) | pq |
| Enlargement, scale factor k, centre the origin | (k, 0; 0, k) | k² |
Written as (a, b; c, d), the first pair is the top row. The general reflection is the one to check on the formula booklet, because θ there is the angle of the MIRROR LINE and the matrix uses 2θ; at θ = 45° it gives (0, 1; 1, 0), the reflection in y = x, which is the quickest way to confirm you have it the right way round. You do not have to memorise any of the others: the first column is where (1, 0) goes and the second column is where (0, 1) goes. Draw the two arrows and you have written the matrix.
Translations are not matrix multiplication. Every matrix here fixes the origin, because a matrix times the zero vector is always zero. A translation moves the origin, so it is a vector you add, not a matrix you multiply by. A transformation that both turns and shifts is written Mx + b.
Let R be a 90° anticlockwise rotation and M a reflection in the x-axis.
R = (0, −1; 1, 0) and M = (1, 0; 0, −1)
Rotate first, then reflect is the matrix MR, because the one that acts first sits closest to the vector:
MR = (0, −1; −1, 0), which is a reflection in y = −x. It sends (1, 0) to (0, −1).
Reflect first, then rotate is RM:
RM = (0, 1; 1, 0), which is a reflection in y = x. It sends (1, 0) to (0, 1).
Two different transformations. Two different mirror lines, perpendicular to each other. And:
det(MR) = −1 and det(RM) = −1
So the determinant is not a name. It tells you two things and no more: its size is the factor every area is multiplied by, and its sign says whether the shape was flipped over. −1 means "same area, flipped", and there are infinitely many transformations that do that. If you want to know which, multiply the matrix by (1, 0) and by (0, 1) and see where they go.
Take the stretch S = (2, 0; 0, 3), with det S = 2 × 3 = 6.
And the sign: det M = −1 for a reflection, and the minus sign is not about area, which is 1 either way. It records that a shape traced anticlockwise comes out traced clockwise. Area uses the size of the determinant, so take the absolute value.
A determinant of 0 is the interesting one. (2, 4; 1, 2) has determinant 2×2 − 4×1 = 0, and it squashes the whole plane onto a single line. Every area becomes 0, nothing can be undone, and the matrix has no inverse. A zero determinant is the warning that information has been destroyed.
Both machines multiply matrices and take determinants directly, which means the only thing left for you to get wrong is the order. Type both orders and look at them.
When you may use it. Applications. A calculator is allowed in every paper, and matrix work is exactly the sort of arithmetic the course expects you to hand over. Enter the matrices, read the product, and spend your time on which order the question described.
The mark people lose. Multiplying in the order the words appear. "Rotate, then reflect" becomes MR, with the rotation on the RIGHT, because it is the one that touches the vector first. Writing RM there gives a reflection in the other line, a wrong image, and a determinant that looks perfectly healthy. Both machines will happily compute the wrong product. The check is to send (1, 0) through by hand and see whether the answer matches the picture.
1. Find the determinant of (2, 0; 0, 3).
2. A triangle of area 4 is transformed by that matrix. What is the new area?
3. What is the determinant of a rotation through 30°?
4. The point (2, 0) is rotated 90° anticlockwise and then reflected in the x-axis. Give the y-coordinate of where it lands.
5. Two different composite matrices both have determinant −1. What does that tell you?
1 markEach transformation's matrix, written correctly.
1 markThe product, in the right order.
1 markThe image, or the determinant, or the area factor, as asked.
The order mark is the one people drop. Write the matrix of the FIRST transformation on the right of the product, every time, and write it down before you reach for the calculator.
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