# Thread: Matrix help

1. ## 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!

2. ## Re: Matrix help

$\displaystyle A=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix}$

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

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

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

According to the question,

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

Therefore you have the following equations,

$\displaystyle 6+y=6+7x \\ 2x+14=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16$

3. ## 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)

4. ## Re: Matrix help Originally Posted by sbhatnagar $\displaystyle A=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix}$

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

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

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

$\displaystyle 6+y=6+7x \\ 2x+4=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16$

5. ## 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)

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