Another Linear Algebra Question

Thread starter mpm
Start date Sep 6, 2005
Tags

Algebra Linear Linear algebra

AI Thread Summary

To prove that if matrix A is nonsingular, then its transpose A^T is also nonsingular, and that (A^T)^{-1} = (A^{-1})^T, one can start from the equation (A^{-1}A)^T = I. Using the hint that (AB)^T = B^T*A^T, the proof can be constructed by applying the properties of transposes and inverses. The discussion emphasizes the importance of understanding these properties in linear algebra. Clarification on the steps involved in the proof is sought, indicating a need for further explanation. This highlights the challenges faced by students in grasping linear algebra concepts.

Sep 6, 2005

mpm

I am having to do a proof on a problem and am not really seeing it for some reason. Maybe it's because I have been doing the homework for so long.

Prove that if A is nonsingular then A^T is nonsingular and

(A^T)^-1 = (A^-1)^T

Hint: (AB)^T = B^T*A^T

I understand the hint, but I can't seem to get an image of the actual problem.

Can anyone help me?

Physics news on Phys.org

Sep 6, 2005

quasar987

Science Advisor

Homework Helper

Gold Member

Start from (A^{-1}A)^T=I.

Thread 'I've come across another fluid pressure problem I don't understand'

Neglect gravity other than that of water; neglect friction. F1=ρgsh=1000*10*1*(1+4)=50000 N; F1=ρgsh=1000*10*0.1*4=4000 N; F3=50000-4000=46000 N?

View full post »

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...

View full post »

Thread 'Understanding Forces and their Vector Components'

I was told forces are vectors so they can be divided into x and y components, but what about the normal force or the force of static friction? do you divide those into x and y components if say you had a block that was lying on an angled surface?

View full post »