1. ## Linear Combinations

A = $\displaystyle $\left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & -1 & 4 \\ -1 & 0 & 1 \end{array} \right)$$,
x = $\displaystyle $\left( \begin{array}{ccc} 1 \\ -2 \\ 3 \end{array} \right)$$,
y = $\displaystyle $\left( \begin{array}{ccc} 3\\ 0\\ 1 \end{array} \right)$$,
z = $\displaystyle $\left( \begin{array}{ccc} 4\\ -2\\ 5 \end{array} \right)$$

I need to compute Ax, Ay, Az as linear combinations of the columns of A.

I'm stuck here, I have tried multiplying x, y and z with A and got:

x = $\displaystyle $\left( \begin{array}{ccc} -1 \\ 17 \\ 2 \end{array} \right)$$,
y = $\displaystyle $\left( \begin{array}{ccc} 7\\ 13\\ -2 \end{array} \right)$$,
z = $\displaystyle $\left( \begin{array}{ccc} 7\\ 34\\ 1 \end{array} \right)$$

Then it asks me to use the answers to compute the product:

$\displaystyle $\left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & -1 & 4 \\ -1 & 0 & 1 \end{array} \right)$$$\displaystyle $\left( \begin{array}{ccc} 1 & 3 & 4 \\ -2 & 0 & -2 \\ 3 & 1 & 5 \end{array} \right)$ Can someone help me out on what to do? I'm really confused... 2. First you should write those results as "Ax= ", "Ay= ", and "Az= ", not "x= ", etc. The columns of A are, of course, \displaystyle \begin{bmatrix}2 \\ 3\\ -1\end{bmatrix}, \displaystyle \begin{bmatrix}3 \\ -1\\ 0\end{bmatrix}, and \displaystyle \begin{bmatrix}1 \\ 4\\ 1\end{bmatrix} You want to write [tex]Ax= \displaystyle \begin{bmatrix}-1 \\ 17\\ 2\end{bmatrix}= \alpha\begin{bmatrix}2 \\ 3\\ -1\end{bmatrix}+ \beta\begin{bmatrix}3 \\ -1\\ 0\end{bmatrix}+ \gamma\begin{bmatrix}1 \\ 4\\ 1\end{bmatrix}$$\displaystyle = \begin{bmatrix}2\alpha+ 3\beta+ \gamma \\ 3\alpha- \beta+ 4\gamma \\ -\alpha+ \gamma\end{bmatrix}$
so you need to solve the three equations $\displaystyle 2\alpha+ 3\beta+ \gamma= -1$, $\displaystyle 3\alpha- \beta+ 4\gamma= 17$, and $\displaystyle -\alpha+ \gamma= 2$ for $\displaystyle \alpha$, $\displaystyle \beta$, and $\displaystyle \gamma$.

For the second problem, you are simply being asked to multiply two matrices. Don't you know how to do that? You seem to have multiplied a matrix and a vector (column matrix). Just multiply the first matrix times each column in the second matrix, giving the three columns of the product matrix.