Matrix equations

Matrix Equations

A matrix equation is an equation of the form Ax = b, where A is a matrix and x and b are vectors. Matrix equations provide a compact way to represent and solve systems of linear equations.

Key Concepts

• Matrix multiplication interpretation: Ax is the linear combination of columns of A with weights from x
• Existence: A solution exists when b lies in the column space of A
• Uniqueness: The solution is unique when A has full column rank (all columns are pivot columns)
• Homogeneous case: Ax = 0 always has the trivial solution x = 0

Properties

• If A is square and invertible, Ax = b has the unique solution x = A⁻¹b
• If A is not invertible, the system may have no solution or infinitely many solutions
• The set of all solutions to Ax = 0 forms the null space of A

Matrix equations unify the treatment of linear systems and connect algebraic computation with geometric concepts like span, linear independence, and subspaces.