C++ :: Widget That Implement Some Matrix Functions Using Data From A Text File

I need to write a widget that will implement some matrix functions using data from a text file. The input data will be spreadsheet like and fully determined with complete rows and columns. I need to do the following,

1. populate a data structure with the input data (double) to create matrix x
2. transpose the input data matrix x to create x'
3. multiply x * x' to create a square matrix x'x
4. take the determinant of x'x

This is pretty standard linear algebra, but not something I have done in cpp before. Implementation such as the best data types to use to store each of the three matrices, how they are sized, and what library functions may be available for the matrix functions like transpose, multiply, determinant, etc.

I will be removing each data row from the initial input to observe the effect on the determinant in case that has any effect on program design.