Question

Consider the following matrix.

Answer 1

I am pondering this matrix, it is quite mysterious ^^

Answer 2

Consider the following matrix.

A =
a b c
d e f
g h i


Suppose that det(A) = 4. Find the determinant of the following matrix.

B =
a + 4g b + 4h c + 4i
-g -h -i
4d 4e 4f

Answer 3

assume square brackets are around the entries in A and B

Answer 4

Okay, so the first matrix has a determinant of aei+bfg+cdh-(ceg+bdi+afh)=4

Answer 5

right

Answer 6

The second matrix would have a determinant of:
-4(a+4g)hf-4(b+4h)id-4(c+4i)ge-(-4(c+4i)hd-4(b+4h)gf-4(a+4g)ie)

Answer 7

okay

Answer 8

Okay nvm all that. There is a much easier way I just remembered.

Answer 9

lol okay

Answer 10

According to inverse matrix row reduction rules, every time you switch two rows, multiply the determinant by -1. Every time you multiply a row by a constant, multiply the determinant by that constant. Every time you add a multiple of one row to another, you don't need to do anything (determinant stays the same).

Answer 11

ahhh i remember that. so it's -16 then?

Answer 12

Yep, sounds about right. Switched rows once, so *(-1), and multiplied a row by 4, so *4. -4*4=-16

Answer 13

*4 *-1, and 1 flip, and one addition of a scaled row

Answer 14

But that last part doesn't affect the determinant's value, right?

Answer 15

as long as youve remembered the rules correctly :)

Answer 16

apparently -16 isnt right. its for 2 marks but when i entered -16, i got 1 out of 2 so something is wrong

Answer 17

abc
def
ghi

flip 2 rows

abc
ghi
def

-1*r2
4*r3

a b c
-g -h -i
4d 4e 4f


add a scaled row to r1

Answer 18

det = 4

flip x -1; det = -4

*4*-1 ; det = 16

Answer 19

Oh haha negative derps. Forgot that r2 got that -1 multiplied to it.

Answer 20

oh okay that makes sense. thanks everyone

Answer 21

Was a good review for me too, gl in the future