For systems of linear equations, row operations — swapping two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another — transform a matrix without changing the solution set of \(Ax = b\). Repeated application produces row echelon form, where each leading entry of a row lies to the right of that of the row above and zero rows sit at the bottom, and the further-normalized reduced row echelon form (RREF), in which each leading 1 is the only nonzero entry in its column. The Rank–Nullity theorem ties everything together: for an \(m \times n\) matrix \(A\), \(\mathrm{rank}(A) + \mathrm{nullity}(A) = n\). Rank is the dimension of the column (or row) space, and nullity is the dimension of the null space, the set of solutions to \(Ax = 0\).
Applications appear throughout the sciences. In probability, a stochastic matrix has nonnegative entries whose rows (or columns) sum to 1; the row-stochastic version serves as the transition matrix of a Markov chain, with entry \(P_{ij}\) giving the probability of moving from state \(i\) to state \(j\). The steady state \(\pi\) is the probability vector satisfying \(\pi P = \pi\) and \(\sum_i \pi_i = 1\) — an eigenvector of \(P^T\) with eigenvalue 1. The Perron–Frobenius theorem guarantees that a positive square matrix has a unique largest real eigenvalue, the Perron root, with a positive eigenvector, and that the spectral radius is a simple eigenvalue. In statistics, the covariance matrix \(\Sigma = E[(X - \mu)(X - \mu)^T]\) of a random vector is symmetric and positive semidefinite, and PCA uses the eigendecomposition of \(\Sigma\): the principal components are eigenvectors of \(\Sigma\), with eigenvalues giving the variance explained along each direction. Symmetric matrices are particularly important in applications because they have real eigenvalues and orthogonal eigenvectors, they arise naturally in quadratic forms and mechanical systems, and they are numerically stable to diagonalize.
Quadratic forms \(Q(x) = x^T A x\), defined by symmetric matrices, classify definiteness through their sign on nonzero vectors. The signature of a symmetric matrix records the count of positive, negative, and zero eigenvalues, while the inertia is the triple \((n_+, n_-, n_0)\) of these counts. Sylvester's law of inertia states that two real symmetric matrices are congruent — that is, \(B = S^T A S\) for some invertible \(S\) — precisely when they share the same inertia. Congruence preserves definiteness but not eigenvalues.
A smooth counterpart to all of this is the matrix exponential \(\exp(A) = \sum_k A^k / k!\), which converges for every square matrix \(A\) and solves the differential equation \(dX/dt = A X\) with \(X(0) = I\). The Jacobian matrix of a vector-valued function \(f: \mathbb{R}^n \to \mathbb{R}^m\) has entries \(J_{ij} = \partial f_i / \partial x_j\), and the determinant of the Jacobian represents the local volume scaling factor of the transformation near a point, neatly linking the geometry of multivariable calculus with the algebra of determinants.