The identity matrix

For two-dimensional column vectors, I = [[1,0],[0,1]]. Multiplying I by (x,y) gives (x,y) because the first row selects x and the second selects y. Multiplying a 2 × 2 matrix by I on either side also returns that matrix. Check the dimensions before multiplying.

The identity transformation fixes every point, including the unit square. A combined transformation may equal I even when its individual steps move points; two reflections in the same line are an example. The fact that a particular point is fixed does not establish that a matrix is I: a reflection fixes all points on its mirror line while moving others.

Worked example

Show that two reflections in the x-axis combine to the identity transformation.

1. The reflection matrix is A = [[1,0],[0,−1]].
2. Multiply A by itself: the diagonal entries become 1 and (−1)² = 1; both off-diagonal entries remain zero.
3. The product is I and therefore fixes every point.

Answer: A² = [[1,0],[0,1]] = I

Revise first: Matrix multiplication.