# How do you know if a matrix does not have an inverse?

**If**the determinant of the

**matrix**is zero, then it will not

**have an inverse**; the

**matrix**is then said to be singular. Only non-singular

**matrices have**inverses. Find the

**inverse**of the

**matrix**A = ( 3 1 4 2 ). result should be the identity

**matrix**I = ( 1 0 0 1 ).

A.

### What is the invertible matrix Theorem?

**Invertible Matrix Theorem**. The

**invertible matrix theorem**is a

**theorem**in linear algebra which gives a series of equivalent conditions for an square

**matrix**to have an

**inverse**. In particular, is

**invertible**if and only if any (and hence, all) of the following hold: 1. The linear transformation is one-to-one.

#### What is the rank of a matrix?

The maximum number of linearly independent vectors in a**matrix**is equal to the number of non-zero rows in its row echelon**matrix**. Therefore, to find the**rank of a matrix**, we simply transform the**matrix**to its row echelon form and count the number of non-zero rows.#### What does it mean for a matrix to be nonsingular?

A square**matrix**that is not**singular**, i.e., one that has a**matrix**inverse.**Nonsingular matrices**are sometimes also called regular**matrices**. A square**matrix**is**nonsingular**iff its determinant is nonzero (Lipschutz 1991, p. 45).#### When a matrix is not invertible?

A**square matrix**that is**not invertible**is called singular or degenerate. A**square matrix**is singular if and only if its determinant is 0. Singular**matrices**are rare in the sense that a**square matrix**randomly selected from a continuous uniform distribution on its entries will almost never be singular.

Updated: 28th October 2019