I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove explicitly and which did not depend on block multiplication was the fact that product of any two $2 \times 2$ upper triangular matrices is an upper triangular matrix.
I had sometimes in past come across this comment by Olga Tausky-Todd that "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." and I am wondering if there is any connection between the two?
Hence, I would like to know what kind of properties about matrices can be proven with block multiplication.