When is a matrix not invertible?

Answer 1

1. if the matrix M is nxm and n is different from m
2. if M is a square matrix (n=m) then the following are equivalent:
-M is not bijective
-Ker(M) is not {0}
-det (M) =0
.

Answer 2

Let A be a square n by n matrix over a field K (for example the field R of real numbers). Then the following statements are equivalent:

A is invertible.
A is row-equivalent to the n-by-n identity matrix In.
A has n pivot positions.
det A ≠ 0.
rank A = n.
The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0})
The equation Ax = b has exactly one solution for each b in Kn.
The columns of A are linearly independent.
The columns of A span Kn (i.e. Col A = Kn).
The columns of A form a basis of Kn.
The linear transformation mapping x to Ax is a bijection from Kn to Kn.
There is an n by n matrix B such that AB = In.
The transpose AT is an invertible matrix.
The matrix times its transpose, is an invertible matrix.
The number 0 is not an eigenvalue of A.

Take any one of these statements. The negation of that statement implies A is NOT invertible.

Answer 3

An n x n matrix is not invertible if the determinant is equal to 0.

Answer 4

when the determinant of an n x n matrix is zero, it is said to be singular and has no inverse