Maths question, matrices

[jak731]

OK before anyone gets in fast the answer is not 42

The question asks for the determinants of matrices A and B, which both come out as 3, it then says explain what you find. I don't have a clue besides saying they just happen to be the same, i know only the bottom row is different so this might have something to do with a decent explanation.

5 -3 3=A
2 -1 1
4 1 2

5 -3 3=B
2 -1 1
6 0 3

Any useful insight appreciated

Edit: they looked a bit neater when i made the post, seems the forum just rounds multiple spaces down to one.

 

[jak731]

OK, so if i wanted to i could find the inverse of both matrices, how would this explain why their determinants are the same?

 

[jak731]

Equal determinants means A and B are similar matrices, which means for some X, A=X'BX (that should be -1, but I can't get a superscript). You might also want to note that if in A you add row 2 to row 3 you get B (which is a transformation that doesn't change the determinant).

Thanks, i didn't spot that row 3 of B was the second and third row of A added together earlier.

I expect the answer you're expected to give is that the determinants are the same because (as Nimble noted) if you add row 2 to row 3 in A, you get B.

I still don't see why adding one row to another results in the determinate being the same, is this true for all matrices or only special cases?

 

[jak731]

Thanks all, i can do the rest of the question now