What kind of transformation is performed by the matrix

𝐴
=
[
1

0

0

−
1
]
?

Question

What kind of transformation is performed by the matrix

𝐴
=
[
1
	
0

0
	
−
1
]
?

BlackTom AI · Accepted Answer

Matrix A = [[1, 0], [0, −1]] maps any vector (x, y) to (x, −y), so it reflects points across the x-axis (flips the y-coordinate).

1. Relevant knowledge points and concepts
- Linear transformations in R^2 are represented by 2×2 matrices.
- The image of the basis vectors and the effect on coordinates determine the geometric transformation.
- Determinant negative implies orientation reversal (reflection), eigenvalues ±1 indicate directions unchanged/reversed.

2. Complete reasoning process, step by step
- Write A explicitly: A = [ [1, 0], [0, −1] ].
- Apply A to a general vector (x, y):
  A(x, y) = (1·x + 0·y, 0·x + (−1)·y) = (x, −y).
- Interpretation: x-coordinate unchanged, y-coordinate negated.

3. Calculations with intermediate steps
- Multiplication: [ [1,0],[0,−1] ] [ x ; y ] = [ 1*x + 0*y ; 0*x + (−1)*y ] = [ x ; −y ].
- Determinant: det(A) = 1·(−1) − 0·0 = −1 (negative → orientation reversing).
- Eigenvalues: det(A − λI) = (1−λ)(−1−λ) = −(1−λ^2) → λ^2 − 1 = 0 → λ = ±1.
  - λ = 1 eigenvector: y = 0 → (1, 0) (x-axis direction unchanged).
  - λ = −1 eigenvector: x = 0 → (0, 1) (y-axis direction reversed).

4. Why the correct answer is correct
- The rule (x, y) → (x, −y) is precisely the reflection across the x-axis: every point keeps its horizontal position but is mirrored vertically. The eigenanalysis shows vectors along the x-axis remain the same, while those along the y-axis are negated, matching a reflection across the x-axis.

5. Why other common transformations are incorrect
- Reflection across the y-axis would give (−x, y) and matrix [−1,0;0,1], not A.
- Rotation matrices have form [cosθ, −sinθ; sinθ, cosθ] and determinant +1, not −1 as here.
- Uniform scaling multiplies both coordinates by the same scalar (both diagonal entries equal), not (1, −1).
- Shear matrices have nonzero off-diagonal entries, which A does not.

6. Conclusion
The matrix A = [[1,0],[0,−1]] performs a reflection across the x-axis, which matches the provided answer: "Reflection in y (across x-axis)".

What kind of transformation is performed by the matrix 𝐴 = [ 1 0 0 − 1 ] ?Single choice

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