Characterization of Invertible Matrices

This lecture covers the characterization of invertible matrices, stating that a square matrix A is invertible if and only if it is equivalent to the identity matrix. The properties of invertible matrices are explored, including having n pivot positions, unique solutions to systems of equations, and linearly independent columns. The lecture also discusses the consequences of matrix multiplication resulting in the identity matrix, the factorization LU, and the properties of unitary matrices. Additionally, it delves into the proof of invertibility, the application of linear transformations associated with invertible matrices, and the significance of triangular matrices in matrix operations.