# Matrix help

• May 20th 2012, 10:56 PM
Hooperoo
Matrix help
Hi guys. Strange question here but i dont know how to solve and its for revision.

let A = 2x2 matrix ( 2 is the top left entry, 1 the top right entry, 7 the bottom left and 4 the bottom right)

B = 2x2 matrix (3 the top left entry, x the top right, y the bottom left and 4 the bottom right)

determine all the possible values for x and y, but such that AB = BA

please help :( this is a revision question for an upcoming exam so i really need to know how to do it. thanks!
• May 21st 2012, 01:44 AM
sbhatnagar
Re: Matrix help
$A=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix}$

$B=\begin{bmatrix}3 & x \\ y & 4\end{bmatrix}$

$AB=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix} \begin{bmatrix}3 & x \\ y & 4\end{bmatrix}=\begin{bmatrix} 6+y & 2x+14 \\ 21 + 4y & 7x+16\end{bmatrix}$

$BA=\begin{bmatrix}3 & x \\ y & 4\end{bmatrix}\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix} =\begin{bmatrix} 6+7x & 3+4x \\ 2y+28 & y+16\end{bmatrix}$

According to the question,

$\begin{bmatrix} 6+y & 2x+14 \\ 21 + 4y & 7x+16\end{bmatrix}=\begin{bmatrix} 6+7x & 3+4x \\ 2y+28 & y+16\end{bmatrix}$

Therefore you have the following equations,

$6+y=6+7x \\ 2x+14=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16$
• May 22nd 2012, 02:27 AM
Hooperoo
Re: Matrix help
But how do i get x and y from there?

I tried one way but i dont know how to find the inverse of a four square (and im not meant to in my course)
• May 22nd 2012, 03:05 AM
Goku
Re: Matrix help
Quote:

Originally Posted by sbhatnagar
$A=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix}$

$B=\begin{bmatrix}3 & x \\ y & 4\end{bmatrix}$

$AB=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix} \begin{bmatrix}3 & x \\ y & 4\end{bmatrix}=\begin{bmatrix} 6+y & 2x+14 \\ 21 + 4y & 7x+16\end{bmatrix}$
$6+y=6+7x \\ 2x+14=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16$

$AB=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix} \begin{bmatrix}3 & x \\ y & 4\end{bmatrix}=\begin{bmatrix} 6+y & 2x+4 \\ 21 + 4y & 7x+16\end{bmatrix}$

$6+y=6+7x \\ 2x+4=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16$
• May 22nd 2012, 03:25 AM
Deveno
Re: Matrix help
you're not meant to find the inverse of anything.

from 6 + y = 6 + 7x, we have:

y = 7x. so if we find out x, we will know y.

from 2x + 4 = 3 + 4x we get:

2x - 1 = 0, so x = 1/2.

we need to check the last two equations for consistency:

21 + 4(7/2) = 21 + 14 = 35
2(7/2) + 28 = 7 + 28 = 35, the 3rd equation is consistent with the first 2.

7(1/2) + 16 = 7/2 + 32/2 = 39/2
7/2 + 16 = 7/2 + 32/2 = 39/9, all 4 equations are consistent.

therefore B can only be:

[3 1/2]
[7/2 4]

(no inverses were harmed in the making of this film)