Matrices Proof> C=A-B, if Ax=Bx where x is nonzero, show C is singular

  • Thread starter Thread starter Luxe
  • Start date Start date
  • Tags Tags
    Matrices Proof
AI Thread Summary
If Ax = Bx for a nonzero vector x, then it follows that Ax - Bx = 0, leading to x(A - B) = 0, which can be rewritten as Cx = 0 where C = A - B. Since x is nonzero and Cx = 0, this indicates that the matrix C must be singular. The discussion confirms that the existence of a nonzero solution to Cx = 0 implies C does not have an inverse. Thus, the conclusion is that C is indeed singular. This proof demonstrates the relationship between the matrices A, B, and C in terms of their singularity.
Luxe
Messages
10
Reaction score
0

Homework Statement


Let A and B be n x n matrices and let C= A - B.
Show that if Ax=Bx, and x does not equal zero, then C must be singular.


Homework Equations





The Attempt at a Solution


Ax-Bx=0
x(A-B)=0
x(C)=0
So, Cx=0

Does that mean C is singular?
 
Physics news on Phys.org
If Cx=0 and x is not the zero vector, then what would C^(-1)(0) be? C0=0 as well. Would it be x or 0? Sure, it means C is singular.
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
View full post »

Similar threads

What Are the Values of A, B, and C for the Given Expression?
Replies
19
Views
2K
Show that ##(x-a)(x-b)=b^2## has real roots- Quadratics
Replies
6
Views
2K
Questions about proof of upper and lower bound theorem for polynomials
Replies
11
Views
2K
Is there a reason Lang uses (x-s)^2 instead of (x+s)^2?
Replies
4
Views
2K
Multiple in logarithm question -- find the variable?
Replies
16
Views
3K
Prove a rational fraction is equal to another
Replies
5
Views
2K
The discriminant of a quadratic and real solutions
Replies
2
Views
1K
Proving Even Integer Coefficients in Quadratic Polynomials - Homework Question
Replies
3
Views
1K
Got stuck due to the inequality not being satisfied
Replies
21
Views
3K
How Do You Solve for X in a Matrix Equation?
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top