I really need help with this question, I do not understand matrices.

Hi Guest!
Ok, this question is about matrix multiplicaton and the dot product (Also called: Inner product or scalar product).

u is a vector, so it has components: $u_1,u_2,...u_n$

Similarly, v is a vector, so it has components: $v_1,v_2,...,v_n$

Now, no need to actually make it in n-dimension. I mean we can, but why not simplify this and assume we are in 2 dimensional space?
That is, n=2
So, u $=$  and v $=$

So our matrix A is:  $A=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}$  (As given, u is the first row and v is the second row)

But we are given an angle and lengths of these vectors. Can we find the "dot product"? 
Yes: $u \dot {} v=u_1v_1+u_2v_2=cos(\theta)*\left || u |\right |*\left || v| \right |=cos(60)*3*2=3$ 

So, we got: $u_1v_1+u_2v_2=3$

Next, let's see what the question really wants:
$Au=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}$ $*\begin{bmatrix} u_1 \\ u_2 \end{bmatrix}$ = $\begin{pmatrix} u_1^2+u_2^2\\ u_1v_1+u_2v_2 \end{pmatrix}$ = $\begin{pmatrix} 3^2\\ 3 \end{pmatrix}=\begin{pmatrix} 9\\ 3 \end{pmatrix}$

Now, in a similar way, find $Av$.

Hope this helps!

The dot product of u and v is equal to ||u||* ||v||*cos 60 =  3. (Prove this before you start using it!)

row_1(Au) = $\mathbf{u \cdot u}$ = 9

row_2(Au) = Dot product of u and v, equaling 3

Au = <9, 3>

Similarly:

row_1(Av) = 3

row_2(Av) = v \cdot v = 4

Av = <3, 4>

There seems to be a notational or transcription error in the question.

The statement that the first row of the matrix A is the vector u implies that u is a row vector.

(Similarly, the implication is that v is a row vector.)

We are not given the dimension of u and v, but assuming that it is two, A will be a 2 by 2 matrix.

In that case, the products Au and Av do not exist, for the products to exist u and v would need to be column vectors.

The question would make sense if we were given that

$\displaystyle \text{row}_{1}(A)=u^{\text{T}},$

(and similarly for v),

or, calculate

$\displaystyle \text{A}u^{\text{T}},$

(If u is a row vector.)