Theory and practice of obtaining triangular, orthogonal and singular expansions of real matrices are considered. It is shown how these decompositions and their underlying transformations are used to solve systems of linear algebraic equations (in particular, poorly conditioned and degenerate), inverse and pseudo-inverse matrices, calculate eigenvalues ​ ​ and singular values, solve linear least squares problems, and some other problems. The presentation of the material is accompanied by specific algorithms and numerical examples.