Introduction to Algorithms by Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest and Clifford Stein : Chapter 28

The authors discuss some common definitions related to matrix, e.g. transpose, vector, column vector, row vector, unit vector, zero matrix, diagonal matrix, square matrix, identity matrix, tridiagonal matrix, upper-triangular matrix, unit upper-triangular matrix, lower-triangular matrix, unit lower-triangular matrix, permutation matrix, symmetric matrix, matrix addition, scalar multiple of a matrix, negative of a matrix, matrix subtraction, matrix multiplication, inner product, outer product, inverse, determinant, positive definite matrix.

The authors then describe Strassen's algorithm for matrix multiplication. The authors then discuss solving systems of linear equations and the use of matrix in solving them. The authors also discuss the concept of LUP decomposition of a matrix A. The authors also discuss forward-substitution and back-substitution to solve lower triangular system. The authors first describe a method in which $P=I_n$ and $A$ is decomposed into two matrices $L$ and $U$. The authors provide an algorithm to do it and proves its running time to be $\Theta(n^3)$. On the basis of this algorithm, the authors provide LUP-decomposition algorithm and proves its running time to be $\Theta(n^3)$.

The authors then prove that running time of matrix inversion and multiplication is of the same order.