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
Additional Details
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.
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
Additional Details
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.