Invertible matrix characteristics

A is invertible

A is row equivalent to the nxn identity matrix

When we say matrix A is row equivalent to the identity matrix, it means that you can transform A into the identity matrix I by applying a sequence of elementary row operations

A has n pivot positions

every row contains a leading entry (pivot) in its reduced row echelon form

The equation Ax=0 has one trivial solution

If the homogeneous equation Ax=0 has only the trivial solution x=0, it means there are no nonzero solutions.

The columns in A form a linearly independent set

A set of columns in a matrix forms a linearly independent set if no column can be written as a linear combination of the others

The linear transformation x→ Ax is one to one

A linear transformation T:ℝn→ℝm defined by T(x)=Ax is one-to-one (injective) if and only if different inputs produce different outputs.

The equation Ax = b has at least one solution for each b

The columns of A span Rn

if the columns of A span the vectors for example a 3×3 matrix spaning x1, x2, and x3

the linear transformation x→ Ax maps Rn → Rn

If the linear transformation T:ℝn→ℝn is defined by T(x)=Ax, then A must be an n×n matrix because the domain and codomain have the same dimension

There is an nxn matrix C such that CA = I

There is an nxn matrix D such that AD = I

A^t is an invertible matrix

ad-bc doesnt equal 0 =

invertible

ad-bc = 0

not invertible