Linear Algebra math problem, need help

nsk11 · **Posted:** Mon, 11 Oct 2010 03:58:57 UTC

Hey i have this math assignment due and i am really stuck. here it is:
Let A be a 3x2 matrix and B be a 2x3 matrix.

If AB is a 3x3 matrix =
8 2 -2
2 5 4
-2 4 5

Show that BA is a 2x2 matrix =
9 0
0 9

Any help would be appreciated!!
Thanks,

Justin · **Joined:** Wed, 21 May 2003 04:27:18 UTC **Posts:** 991

nsk11 wrote:

Hey i have this math assignment due and i am really stuck. here it is:
Let A be a 3x2 matrix and B be a 2x3 matrix.

If AB is a 3x3 matrix =
8 2 -2
2 5 4
-2 4 5

Show that BA is a 2x2 matrix =
9 0
0 9

Any help would be appreciated!!
Thanks,

The matrix AB is not invertible, so this limits what you can do with it....were you not given any more information??

nsk11 · **Posted:** Mon, 11 Oct 2010 16:58:38 UTC

no my teacher said this is all i need

my attempt:

Brute force, using a system of 13 equations should not be the way to solve this question. This is more of a proof like problem.

Two things noted:
(1) AB is symmetric -- this should have something to do with the problem.
(2) Trace BA = Trace AB = 18 ( trace = sum of diagonals -- must be equal)

Also, by adding a row or column of zeros you can get 3 x 3 matrices, somewhat easier to work with. All determinants are zero so you can't invert them however.

outermeasure · **Posted:** Mon, 11 Oct 2010 17:00:54 UTC

nsk11 wrote:

Hey i have this math assignment due and i am really stuck. here it is:
Let A be a 3x2 matrix and B be a 2x3 matrix.

If AB is a 3x3 matrix =
8 2 -2
2 5 4
-2 4 5

Show that BA is a 2x2 matrix =
9 0
0 9

Any help would be appreciated!!
Thanks,

Hint: Changing

to $P^{-1}A$ and

to

at the same time has the effect of changing

to $P^{-1}(AB)P$ and leaving BA unchanged. Similarly, changing

to

and

to $Q^{-1}B$ has the effect ....

nsk11 · **Posted:** Mon, 11 Oct 2010 17:37:01 UTC

Changing A to P^-1 * A and B to BP at the same time has the effect of changing AB to (P^-1)(AB)P and leaving BA unchanged. Similarly, changing A to AQ and B to (Q^-1)B has the effect (Q^-1)(BA)Q and leaving AB unchanged.

I understand that, where should i go from there?

outermeasure · **Posted:** Mon, 11 Oct 2010 18:13:06 UTC

nsk11 wrote:

Changing A to P^-1 * A and B to BP at the same time has the effect of changing AB to (P^-1)(AB)P and leaving BA unchanged. Similarly, changing A to AQ and B to (Q^-1)B has the effect (Q^-1)(BA)Q and leaving AB unchanged.

I understand that, where should i go from there?

Maybe that wasn't that great a hint. It is a simplifying step if you want to brute-force your way.

For a more elegant solution --- Do you know how the minimal polynomials of ST and TS, where S, T are linear maps $S\colon V_1\to V_2$ and $T\colon V_2\to V_1$ , relate to each other?

nsk11 · **Posted:** Mon, 11 Oct 2010 18:40:40 UTC

No not really, i know what a linear map is, but i don't see how it is applicable to this problem

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Linear Algebra math problem, need help

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