# Two matrix problems

• Dec 3rd 2012, 03:03 PM
billb91
Two matrix problems
Hey guys,

I have two matrix problems that I'm a bit unsure about. The first one I think I have the right answer, but just want to be sure, and the second, I think the same applications as the first problem apply, but I'm not entirely sure how to complete it. Here are the two problems:

Attachment 26041 (it didn't post full size so zoom in if you're having trouble making out the numbers)

For the first, my answer was [x y
0 x]

because the diagonal 1s are a property of a unit matrix. Then I'd assume the 0 will remain a zero, and I set the 1 equal to y because I really didn't know what else to do.

Now for the second, I'd assume the unit matrix idea would be the same as in problem one. However, I'm really not sure what else to do with that one. Any help with that would be very much appreciated.

Thanks to anybody who offers help!
• Dec 3rd 2012, 03:53 PM
topsquark
Re: Two matrix problems
Quote:

Originally Posted by billb91
Hey guys,

I have two matrix problems that I'm a bit unsure about. The first one I think I have the right answer, but just want to be sure, and the second, I think the same applications as the first problem apply, but I'm not entirely sure how to complete it. Here are the two problems:

Attachment 26041 (it didn't post full size so zoom in if you're having trouble making out the numbers)

For the first, my answer was [x y
0 x]

because the diagonal 1s are a property of a unit matrix. Then I'd assume the 0 will remain a zero, and I set the 1 equal to y because I really didn't know what else to do.

Now for the second, I'd assume the unit matrix idea would be the same as in problem one. However, I'm really not sure what else to do with that one. Any help with that would be very much appreciated.

Thanks to anybody who offers help!

Your answer to the first part is correct. One way to do this is simply to do each product and make them equal:
$\displaystyle \left [ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right ] \cdot \left [ \begin{matrix} x & y \\ z & w \end{matrix} \right ] = \left [ \begin{matrix} x + z & y + w \\z & w \end{matrix} \right ]$

$\displaystyle \left [ \begin{matrix} x & y \\ z & w \end{matrix} \right ] \cdot \left [ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right ] = \left [ \begin{matrix} x & x + y \\z & z + w \end{matrix} \right ]$

Now just equate the appropriate matrix values and you'll easily derive your answer.

As for the second problem, just multiply it out. I did up to n = 3 and the pattern is obvious. You can try a proof by induction as you like, but it's just as easy to show it case by case.

-Dan
• Dec 3rd 2012, 05:07 PM
billb91
Re: Two matrix problems
Thanks for the help/confirmation! :)