I'm kinda confused in the follow exercise:
Find the correct answer:
A= Matrix 15x15
a) A is invertible and there is a partition with at least 4 blocks which turns this matrix into a block diagonal matrix
b) A is not invertible and there is a partition with at least 4 blocks which turns this matrix into a block diagonal matrix
c) A is invertible and there is a partition with at least 4 blocks which turns this matrix into a block upper triangular matrix
d) A is not invertible and there is a partition with at least 4 blocks which turns this matrix into a block upper triangular matrix
e) A is invertible and there is a partition with at least 4 blocks which turns this matrix into a block lower triangular matrix
f) A is not invertible and there is a partition with at least 4 blocks which turns this matrix into a block lower triangular matrix
I know I can solve this exercise by using the Gaussian elimination method and finding a triangular matrix, but it looks like it's supposed to be done in a easier way, Gauss is pretty long, any different solutions here?
It is easy to see, that you can turn $A$ in an upper block triangle matrix with at least 4 blocks, just take the blocks on the diagonal (either size 1 (rows 1-5,10-12), 2 (rows 13 and 14) or 4 (rows 6-9)).
Now our matrix is invertible if all blocks on the diagonal are invertible, this is due to the fact that the determinant of an upper block matrix is the product of the determinants of the blocks on the diagonal.
As all blocks on the diagonal are invertible because they are either a block of size 1 not zero, they are a lower triangular matrix with no zeros on the diagonal or they are $\left(\begin{matrix}1&2\\ 3 & 4\end{matrix}\right)$ which is invertible (Or one can calculate their determinants).
Their determinants are thereby nonzero, so the determinate of $A$ is nonzero and $A$ is invertible.
So the right answer is c)