Represent the following system of equation in the Y = AX + B form:

Question

$$y_1 = 4x_1 + x_2 + x_3 + 9x_4 + 5$$ $$y_2 = x_1 - x_2 + 6x_3 - x_4 + 1$$ $$y_3 = -2x_1 + 3x_2 - x_3 + 2x_4 - 19$$

What do we need to do? Should we take all on one side and keep the 5 on the other side?

Answer 1

As you just just have one equation, you may set

$$(y_1)=\begin{pmatrix}4 &1 &1 &9\end{pmatrix}\cdot\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}+(5)$$

Note that here $Y=(y_1)$ is a one-dimensional vector(a $1\times 1$ dimensional matrix), $A=\begin{pmatrix}4 &1 &1 &9\end{pmatrix}$ is a $1\times 4$-dimensional matrix, $X=\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}$ is a $4\times 1$-dimensional matrix and $B=(5)$ is a $1\times 1$ dimensional matrix.

The vertical dimensions of $Y,B$ and of $A$ of course depend on the number of equations and the vertical dimension(horizontal dimension) of $A$ depend on the number of variables used.

EDIT: Corresponding to your edit, we may change the above presentation to

$$\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}=\begin{pmatrix}4 &1 &1 &9\\ 1 &-1 &6 &-1\\ -2 &3 &-1 &2\end{pmatrix}\cdot\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}+\begin{pmatrix}5\\ 1\\ -19\end{pmatrix}$$

As a training exercise, lay out the laws of matrix multiplication and matrix addition out in front of you and write some random systems of equations and try to convert them. This really helps to give you an intuition of the underlying matrix operations.