# Multiplying a matrix by its transpose

1. Jul 14, 2012

### g.lemaitre

1. The problem statement, all variables and given/known data

I don't see how you multiply a matrix by its transpose. If a matrix is 3 x 2 then its transpose is 2 x 3. I thought you couldn't multiply matrices unless they have the same rows and columns.

2. Jul 14, 2012

### Pranav-Arora

Re: multiplying a matrix by it's transpose

For matrices to be multiplied, the condition is that the number of columns of the first matrix should be equal to the number of rows of the other matrix.

3. Jul 14, 2012

### genericusrnme

Re: multiplying a matrix by it's transpose

Nope, if a matrix A is n x m and B is m x l then AB is defined

If you do the same procedure of matric multiplication you'll see that multiplying a 3 x 2 and a 2 x 3 matrix gives you a 3 x 3 matrix of rank at most 2
If you multiply 2 x 3 by 3 x 2 you'll get a 2 x 2 matrix with rank, also (and obviously), at most 2

In terms of components if A = BC, where B is n x m and C is m x l, then

$A_{i,k} = \sum_{j=1}^m B_{i,j} C_{j,k}$