Matrix Transformation Inversion?

📌 CONCEPT: The transformation represented by a matrix T is its own inverse, meaning if T is applied twice in succession, the resulting transformation is the identity transformation, which leaves the original shape unchanged.

📐 RULE / FORMULA: For any matrix T, if T^2 represents the transformation resulting from applying T twice in succession, then T^2 = I, where I is the identity matrix.

💡 WORKED EXAMPLE: Consider a 90° clockwise rotation matrix T = [ [0, -1], [1, 0] ]. To find T^2, we multiply T by itself: T^2 = [ [0, -1], [1, 0] ] * [ [0, -1], [1, 0] ] = [ [ -1, 0], [ 0, -1] ]. However, this is the matrix for a 90° counter-clockwise rotation, which when composed with itself gives the identity transformation I.

⚠️ COMMON MISTAKE: Students often mistakenly believe that applying the same matrix transformation twice will result in the original shape, when in fact it will only result in the same shape if the transformation is the identity transformation, otherwise the resulting shape will be different.