Question

Suppose A is a 4 x 2 matrix, B is an r x s matrix and C is a 4 x 2matrix. if (AB) ^t C is defined then R =? and S =?

Answer 1

I'm not sure what you're asking. Are you saying:\[(AB)^T=C\]and we need to find out the dimensions of B?

Answer 2

this is a multiple choice but I don't understand how they work it out. if it is defined by)\[(AB) ^{T} C\]

Answer 3

the answers can be
a) r = 2, s= any positive integer
b) r= 4, s = 2
C) r=2, s = 4
d)r=s=2

Answer 4

I see, so they're saying that matrix is defined, so then the rows and columns have to line up in multiplication.

Since we know A and C are 4x2 matrices, they look like this: \[A=\left[\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22}\\a_{31} & a_{32} \\ a_{41} & a_{42} \end{matrix}\right]\]

So this is getting right multiplied by another matrix, and that matrix must have 2 rows exactly, otherwise it won't multiply with this one. How many columns it has is completely unknown.

A better way of putting it is any time you multiply two matrices, the "inside" indices have to match up, and the matrix you get out of it has the "outside"indices which is true here, see x by y times an y by z matrix makes an x by z \[F_{xy}G_{yz}=H_{xz}\]

So now after you've multiplied that last matrix, you have to transpose it. All that means to use is the number of rows become columns and columns becomes rows, right? So now apply this rule i just showed you with that matrix we've made multiplied by C. It's sort of a lot to keep track of, but try your best and write down as much as you can so you don't have to stress out about juggling it around in your brain and you can focus on the problem better.

Answer 5

For A and B to multiply, it means that the columns of A must equal the rows of B. Thus \(r=2\).

Answer 6

The dimensions of AB will be the rows of A by the columns of B. So we have a 4 by s matrix.

Answer 7

The transpose will be s by 4.

Answer 8

strange though, this doesn't put any limit on s... maybe I mixed them up.

Answer 9

@wio That's right. Look at the multiple choice questions she has, it's an option.