Why Is the Transpose of a Matrix Important in Linear Algebra?

[matqkks]

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.

 

[Fredrik]

I think the main reason why the transpose is useful is that the standard inner product on the vector space of n×1 matrices is $\langle x,y\rangle=x^Ty$. This implies that a rotation R must satisfy $R^TR=I$.

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.

 

[kdbnlin78]

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.