Invertible Matrix Theorem
Let A be a square n Ă— n matrix. Then the following are equivalent:
(i) A is an invertible matrix.
(ii) A is row equivalent to the n Ă— n identity matrix.
(iii) A has n pivots positions.
(iv) The equation Ax = 0 has only the trivial solution.
(v) The columns of A form a linearly independent set.
(vi) The linear transformation x ↦ Ax is one-to-one.
(vii) The equation Ax = b has at least one solution for each b in Rn.
(viii) The columns of A span Rn.
(ix) The linear transformation x ↦ Ax maps Rn onto Rn.
(x) There is an n Ă— n matrix C such that CA = I.
(xi) There is an n Ă— n matrix D such that AD = I.
(xii) A^T is an invertible matrix.
Echelon form
A rectangular matrix is in echelon form (or row echelon form) if it has the following properties:
(i) All nonzero rows are above any rows of all zeros.
(ii) Each (non-zero) leading entry of a row is in a column to the right of the leading entry of the row above it.
(iii) All entries in a column below a leading entry are zeros.
Reduced Row Echelon Form
If a matrix in echelon form satisfies the following additional conditions, then it is in
reduced echelon form (or reduced row echelon form):
(iv) The leading entry in each nonzero row is 1.
(v) Each leading 1 is the only nonzero entry in its column.
Homogenous Linear System
A system of linear equations is said to be homogeneous if it can be written in the form
Ax = 0, where A is an m Ă— n matrix and 0 is the zero vector in Rm.
Such a system Ax = 0 always has at least one solution, namely, x = 0 (this zero vector is in Rn), called the trivial solution.
The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.