First A, then B → the single matrix is BA: Imagine a robot that first rotates a part (matrix A), then enlarges it (matrix B). To get one matrix that does both, multiply: apply A first, then B, so the point goes
image .
The right-most matrix acts first. Because matrix multiplication is not commutative ( in general), the order matters — doing A-then-B usually differs from B-then-A.
IB-style question — single combined matrix
A graphics pipeline first reflects a sprite in the x-axis using , then rotates it 90° anticlockwise using .
Find the single matrix that does both.
Step by step
- “Reflect then rotate” means rotate-matrix on the LEFT: compute BA.
- Row 1: (0·1 + (−1)·0, 0·0 + (−1)·(−1)) = (0, 1).
- Row 2: (1·1 + 0·0, 1·0 + 0·(−1)) = (1, 0).
Final answer
The single matrix is — a reflection in y = x. (On a GDC: enter A and B and compute B×A.)
Order is everything: Swapping the order gives here — a different transformation (a reflection in the line , since ). Always write the matrix of the transformation done last on the left.
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|det| scales area; a negative det flips the shape: For the determinant is .
When M transforms a shape, the area is multiplied by :
new area old area.
The sign carries meaning too: keeps the orientation; means the shape was reflected (turned over).
IB-style question — area after a transformation
A field is mapped to a region of area 12 km². It is transformed by .
Find the area of the image region.
Step by step
- Compute the determinant ad − bc.
- The area scale factor is |det M| = 10.
Final answer
The image has area 120 km². (Since det = 10 > 0, the orientation is preserved — the region isn't flipped.)
IB-style question — what det tells you
A triangle of area 5 is transformed by .
Find the image area and state what the sign of the determinant tells you.
Step by step
- Determinant.
- Area scale factor is the modulus.
- Interpret the negative sign.
Final answer
Image area 15; the negative determinant means the triangle has been flipped over (its orientation is reversed).
IB-style question — rotation × enlargement
A transformation has matrix M = (3, −3; 3, 3) (top row 3, −3).
Show that M is an enlargement combined with a rotation, and state the scale factor and the angle.
Step by step
- An enlargement (scale k) with a rotation (angle θ) has matrix k(cos θ, −sin θ; sin θ, cos θ), and its determinant is k². So k = √(det M).
- Divide M by k to expose the rotation matrix.
- Compare with (cos θ, −sin θ; sin θ, cos θ): cos θ = sin θ = 1/√2.
Final answer
M is an enlargement of scale factor 3√2 ≈ 4.24 together with a 45° anticlockwise rotation about O.
det = 0 means collapse: If the area scale factor is 0 — the whole plane is squashed onto a line (or a point), so the shape collapses and the transformation can't be undone (no inverse exists).