When taking the inverse of a matrix, does the matrix have to be a square matrix in order for it to be invertible?

Question

jim_thompson5910 · Answer

Yes the matrix A must be square so that you can compute

$$\Large A*A^{-1}$$
and 
$$\Large A^{-1}*A$$

which both will lead to the identity matrix $\LARGE I$

jim_thompson5910 · Answer

this is assuming that the inverse exists

anj123 · Answer

So the reason why it has to be square is so that it is commutative?
Following the rule AA^-1 = I  = A^-1A

jim_thompson5910 · Answer

https://people.richland.edu/james/lecture/m116/matrices/inverses.html snapshot of the page http://prntscr.com/bdq652

anj123 · Answer

Is it possible in any situation at all, to take the inverse of a matrix that is not square?

jim_thompson5910 · Answer

Let's assume matrix A wasn't square. Let's say matrix A has the dimensions 3x5, so matrix A has 3 rows and 5 columns

Let matrix A be invertible and let's assume matrix B is the inverse of matrix A

so that means
A*B = I
B*A = I

But we have a problem

A*B is NOT defined if matrix B was of the size 3x5, or 5x3 or even 3x3. If B was 5x5 then it would work BUT it would not work with B*A. So to make all the dimensions match up, we need A and B to both be square matrices. We need them to have the same number of rows and columns.

jim_thompson5910 · Answer

http://people.revoledu.com/kardi/tutorial/LinearAlgebra/MatrixGeneralizedInverse.html snapshot http://prntscr.com/bdq7db that is basically saying technically it is possible, but it's not a full inverse. It would be "half" an inverse so to speak. Either a left-inverse or right-inverse

samigupta8 · Answer

Can you find out determinant of a matrix which is not square?

anj123 · Answer

No, in order to take determinant of a matrix, the matrix must be square.

samigupta8 · Answer

You gave the ans yourself... When we can't evaluate the determinant then we don't talk about the inverse of the matrix