# Math Help - Transpose of matraces

1. ## Transpose of matraces

Hey guys, I understand matrices and their tranpose but I have this question which is just confusing... maybe it's because it's late but anyway it reads:

Given the two vectors: A = [1, 2] B = [3, 4] calculate:

A BT and B AT

So I did this

A = [1, 2]
B =
[3]
[4]

But I don't get what to do next... I can't multiple these can I?

P.S I really must learn how to insert the proper mathematical code to format my questions correctly

2. ## Re: Transpose of matraces

Originally Posted by uperkurk
But I don't get what to do next... I can't multiple these can I?
Why can't you? In both cases you have a 1x2 matrix times a 2x1 matrix. So your results should be 1x1 matrices.

3. ## Re: Transpose of matraces

Nevermind I just worked it out. I was checking my results using a website that has a calculator for matraces. But it only went from 2x2 - 1x3 - 3x3 so I just assumed 1x2 and 1x1 did not exist...

So I got this as my answer

ABT I got 11
BAT I got 11

ATB I got 11
BTA I got 11

Which were the four questions I had to work out so for a conclusion I just wrote, not matter which way you switch them around you'll always get the same answer?

4. ## Re: Transpose of matraces

Originally Posted by uperkurk
ATB I got 11
BTA I got 11
Check these again. In this case you have a 2x1 times a 1x2. You should, therefore, get a 2x2 result. In general, when multiplying an $m\times p$ matrix with a $p\times n$ matrix, the result will be an $m\times n$ matrix.

5. ## Re: Transpose of matraces

Oh... now I think I rushed through them too fast...

For ATB I'm getting a 2x1 matrix of

7
14

and for BTA I'm getting a matrix of

9
12

is this now correct?

6. ## Re: Transpose of matraces

$A^{\textrm T}B = \left[\begin{array}{c} 1 \\ 2\end{array}\right]\left[\begin{array}{cc}3 & 4\end{array}\right] = \left[\begin{array}{cc}1\times3 & 1\times4 \\ 2\times3 & 2\times4\end{array}\right] = \left[\begin{array}{cc}3 & 4\\6 & 8\end{array}\right]$

7. ## Re: Transpose of matraces

What?! I thought when doing matraces you had to multiply and then add...

1 x 3 + 1 x 4 = 7
2 x 3 + 2 x 4 = 14

Then what the hell have I been learning LOL

8. ## Re: Transpose of matraces

Originally Posted by uperkurk
What?! I thought when doing matraces you had to multiple and then add...

1 x 3 + 1 x 4 = 7
2 x 3 + 2 x 4 = 14

Then what the hell have I been learning LOL
If you multiply two matrices $\mathbf A$ and $\mathbf B$, each entry $\left[\mathbf{AB}\right]_{i,j}$ of the resulting product is found by multiplying (taking the dot product of) the $i^\mathrm{th}$ row of $\mathbf A$ by the $j^\mathrm{th}$ column of $\mathbf B$.

We have a 2x1 matrix times a 1x2 matrix, so we should expect our answer to be 2x2:

$\left[\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]$.

So, using the above definition of matrix multiplication, $a_{11}$ is given by the dot product of the first row of $\mathbf A$ and the first column of $\mathbf B$: $a_{11} = 1\cdot 3 = 3$. And so on.

Since the rows of $\mathbf A$ and the columns of $\mathbf B$ have only one entry each, there is no addition to be done.

9. ## Re: Transpose of matraces

ok so I see what you did there, no addition in those cases

Thanks