ATAR Notes: Forum

Uni Stuff => Science => Faculties => Mathematics => Topic started by: QuantumJG on January 04, 2010, 05:05:44 pm

Title: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 04, 2010, 05:05:44 pm: Show that if A is an m x n matrix and A(BA) is defined, then B is an n x m matrix.

Firstly let the size of B = y x z

if A(BA) is defined, then BA is defined => z = m

therefore the size of matrix BA is y x n, so for A(BA) to be defined => y = n

=> size of B is n x m.

Is that an adequate proof?
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: jimmy999 on January 04, 2010, 05:54:51 pm: If you wanted to extend it you could say the number of columns of the first one must equal the number of rows of the second one. Something like that. I'm not exactly sure how detailed proofs need to be
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 04, 2010, 06:11:35 pm: Let matrix $A = \left[ {\begin{array}{cc} a & b \\ c & d \\\end{array} } \right]$ , be a matrix that represents all 2 x 2 matrices

Then AB = BA when B is either

$\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} } \right]$ or $\left[ {\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} } \right]$

Show that matrix A which satisfies the above must be a scalar matrix.
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 04, 2010, 06:22:07 pm: A scalar matrix is a matrix where all diagonal entries are equal, but I'm confused as to how to show A is a scalar matrix.
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: Voltaire on January 04, 2010, 09:35:16 pm: why are you reviewing the intro to matrixies stuff..?

you should do like change of basis, orthonormal set's, then diagonalization and conics.
you gotta understand how it all links up etc, dont be doing these isolated defition type Q's
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 04, 2010, 10:54:07 pm: Quote from: Voltaire on January 04, 2010, 09:35:16 pm
why are you reviewing the intro to matrixies stuff..?

you should do like change of basis, orthonormal set's, then diagonalization and conics.
you gotta understand how it all links up etc, dont be doing these isolated defition type Q's
Quote from: Voltaire on January 04, 2010, 09:35:16 pm
why are you reviewing the intro to matrixies stuff..?

you should do like change of basis, orthonormal set's, then diagonalization and conics.
you gotta understand how it all links up etc, dont be doing these isolated defition type Q's

Lol. I have already done that.

I'm just going through some proof questions I didn't get during the semester.
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 05, 2010, 07:14:41 pm: This question is really annoying me!

Let A be a square matrix satisfying A² = A and let matrix B be a square matrix the same size as A.

Show that (AB - ABA)² = 0 (obviously the 0 matrix)

The only thing I can think of is trying to prove AB = ABA, which would be easy if ABA = AAB = A²B = AB.
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: zzdfa on January 05, 2010, 07:25:12 pm: (AB-ABA)^2
=(AB-ABA)(AB-ABA)
=AB(AB-ABA)-(ABA)(AB-ABA) matrix multiplication is distributive
=ABAB-ABABA-ABAAB+ABAABA and again
=ABAB-ABABA-ABAB+ABABA A^2=A
=0
sometimes you just have to get your hands dirty.
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: zzdfa on January 05, 2010, 07:36:25 pm: Quote from: QuantumJG on January 04, 2010, 06:11:35 pm
Let matrix $A = \left[ {\begin{array}{cc} a & b \\ c & d \\\end{array} } \right]$ , be a matrix that represents all 2 x 2 matrices

Then AB = BA when B is either

$\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} } \right]$ or $\left[ {\begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} } \right]$

Show that matrix A which satisfies the above must be a scalar matrix.

Let $B_1$ be the first matrix and $B_2$ be the second.

then you have 2 equations:

$AB_1=B_1 A$

$AB_2 = B_2 A$

Get those equations in terms of a,b,c,d (do the multiplication e.g. $A B_1=\left[ {\begin{array}{cc} a & 0 \\ c & 0 \\ \end{array} } \right]<br />$ )
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 05, 2010, 07:43:02 pm: Quote from: zzdfa on January 05, 2010, 07:25:12 pm
(AB-ABA)^2
=(AB-ABA)(AB-ABA)
=AB(AB-ABA)-(ABA)(AB-ABA) matrix multiplication is distributive
=ABAB-ABABA-ABAAB+ABAABA and again
=ABAB-ABABA-ABAB+ABABA A^2=A
=0
sometimes you just have to get your hands dirty.

Thanks.

I suck at matrix proofs. :(
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: QuantumJG on January 10, 2010, 05:08:07 pm: How can you do this distributivity proof?

(αa + βb) x c =αa x c + βb x c

the only thing I can think of is setting,

a = (a₁, a₂, a₃)

b = (b₁, b₂, b₃)

c = (c₁, c₂, c₃)

and showing that the RHS = LHS

is there a neater way?
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: humph on January 10, 2010, 11:25:35 pm: Depends what you've already proved/can assume beforehand, but in general no, that's the most direct (albeit lengthy) way.
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: /0 on January 10, 2010, 11:41:13 pm: Here is a proof for coplanar vectors, from Introduction to Electrodynamics - Griffiths. It also includes the source for the general case. (Which, unfortunately, I don't have)
Title: Re: QuantumJG's first year uni maths revision thread.
Post by: kamil9876 on January 11, 2010, 01:41:21 am: I like that geometry :P

I agree with humph, was just about to say that it depends what you have proven earlier/take as the definition of cross product. Like for instance if you know the linearity (in a single row) property of determinants and know how determinants relate to it, this becomes trivial.