
#1
Jul2111, 06:04 AM

P: 150

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. 



#2
Jul2111, 06:16 AM

Emeritus
Sci Advisor
PF Gold
P: 9,014

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 [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. 



#3
Jul2111, 08:45 AM

P: 34

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. 


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