Involutory Matrix

comment down below the only matrix that is both idempotent and involutory. 00:00 – Introduction to Involutory Matrices 00:13 – Definition: When is a matrix involutory? ($A^2 = I$) 00:38 – Examples of Involutory Matrices (Identity, $-I$) 01:10 – Geometric Interpretation: Reflection through $y=x$ 03:44 – Relation between Projection and Reflection 06:05 – Eigen Space and Dimensions (Eigen Value 1) 08:11 – Algebra of Involutory Matrices 08:24 – Addition and Subtraction Conditions 10:18 – Scalar Multiplication ($k = \pm 1$) 11:51 – Multiplication of Two Involutory Matrices (Commutativity) 12:24 – Transpose and Similarity of Involutory Matrices 14:10 – Inverse of an Involutory Matrix (Self-Inverse) 14:58 – Powers of an Involutory Matrix (Even vs. Odd) 16:46 – Result 1: Annihilating Polynomial ($x^2 - 1 = 0$) 17:54 – Result 2: Minimal Polynomial and Diagonalizability 20:11 – Result 3: Eigen Values, AM, and GM 21:31 – Possible Eigen Values ($\pm 1$) 24:34 – Result 4: Trace of an Involutory Matrix 26:43 – Result 5: Determinant Calculation 27:37 – Result 6: Rank of Involutory Matrices 28:13 – Summary of Key Results & Conclusions