Why is the transpose of a matrix important? To find the inverse by cofactors we need the transpose but I would never find the inverse of a matrix by using cofactors.
I think the main reason why the transpose is useful is that the standard inner product on the vector space of nĂ—1 matrices is [itex]\langle x,y\rangle=x^Ty[/itex]. This implies that a rotation R must satisfy [itex]R^TR=I[/itex]. I think that cofactor stuff is sometimes useful in proofs, but you're right that if you just want to find the inverse of a given matrix, there are better ways to do it.
There are of course many ways to invert a matrix but thie is not the only use for the transpose. Systems of linear equations can be reformulated into matrix systems by looking at the equation xAx^{T} = b where x is a n x 1 column vector with entries {x_{1},...,x_{n}} and Z is a square matrix n x n with entries (for real valued equations, say) in /mathbb{R}. The matrix b is then also an $n x 1$ column matrix of numbers in /mathbb{R} too.