Matrix Homework: Solving for B in Statement 5

[athrun200]

Homework Statement

See question 5

Homework Equations

The Attempt at a Solution

For part a, it is very easy.
Multiply the inverse of A 2 times on both side, we can see the B=inverse of A.
i.e. The required B is inverse of A, then the proof is finished.

But how about part b?
It seems it is the same part a.

Is part b also correct?

 

[HallsofIvy]

Part b is quite different from part a- and the difference is important to learn. Mathematics must be very very precise in its wording- unlike science we don't have observations and experiments to fall back on. In other words, we can't just look at the real world- words are everything!

In part a it ask if, given a non-singular matrix A, there exist a matrix B such that AB^2= A. You are right- just multiply, on the left, on both sides by A^{-1}, which exists because A is non-singular, and the equation becomes AB= I. Yes, B exists and is the inverse of A.

In part B, it asks if there exists a matrix B such that, for any non-singular matrix, A, A^2B= A. "Any" is the crucial word there. Is there a single matrix B that is the inverse of all invertible matrices?

 

[athrun200]

Part b is quite different from part a- and the difference is important to learn. Mathematics must be very very precise in its wording- unlike science we don't have observations and experiments to fall back on. In other words, we can't just look at the real world- words are everything!

In part a it ask if, given a non-singular matrix A, there exist a matrix B such that AB^2= A. You are right- just multiply, on the left, on both sides by A^{-1}, which exists because A is non-singular, and the equation becomes AB= I. Yes, B exists and is the inverse of A.

In part B, it asks if there exists a matrix B such that, for any non-singular matrix, A, A^2B= A. "Any" is the crucial word there. Is there a single matrix B that is the inverse of all invertible matrices?

Well, after listening to your explanation, I know part b is obvious wrong.
However, I wonder how to write it out.

 

[HallsofIvy]

"No, there does not exist a single matrix, B, such that A^2B= A for all non-singular matrices, A."

 

[athrun200]

"No, there does not exist a single matrix, B, such that A^2B= A for all non-singular matrices, A."

Oh, this is the prove?

 

[athrun200]

Let me try for part d.

Since A is non-singular, A^{-1} exists.
So \vec{x}=A^{-1}\vec{y} exists.

So, there exists \vec{x} s.t. A\vec{x}=\vec{y}

Again, how to disprove part c?
By simply saying NO, there doesn't exist?