How can a matrix have no inverse

Does it mean that a matrix with no inverse cannot be reduced on the LHS to the identity matrix? Yes it does mean that a matrix with no inverse cant be reduced to identity matrix in the elimination. You can easily check it out subtract two times the 2nd row from the third row you will end up in a all zero third row. The reason behind this is that let a non invertible matrix be reduced to the identity matrix by a series of row operation.

We know each row operation can be thought of as a multiplication of elementary matrices from left,so we have after the row operations,. On the right side we have all the matrices invertible which implies that A must be invertible. This is a contradiction. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero. The inverse of a square matrix A, denoted by A -1 , is the matrix so that the product of A and A -1 is the Identity matrix. The identity matrix that results will be the same size as the matrix A. Wow, there's a lot of similarities there between real numbers and matrices. That's good, right - you don't want it to be something completely different.

There are a couple of exceptions, though. Remember, "There is no Matrix Division! A square matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular. Step 3, find the determinant and divide every element by that. The determinant is the product of the elements on the main diagonal minus the product of the elements off the main diagonal.

We divide every element by Now, you're saying, wait a minute - you said there was no matrix division. We cannot go any further! This matrix has no Inverse. Such a matrix is called "Singular", which only happens when the determinant is zero. And it makes sense There needs to be something to set them apart. Imagine we can't divide by numbers But what if we multiply both sides by A -1?

The calculations are done by computer, but the people must understand the formulas.