Lets write

\(A=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} X=\begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\end{bmatrix}\)

Matrix A , which has the dimensions 4 x 4 , is the matrix of ciefficients of the system , X , which has the dimensions 4 x1 is the matrix of unknowns.

\(AX=0\)

\(\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\end{bmatrix}=\begin{bmatrix}0\\ 0\\0\\ 0\end{bmatrix}\)

If X is a solution of SX=0 , then so is c_1X for any constant c_1 . compute the product AX:

\(AX=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}1\\ -1\\ -1\\1\end{bmatrix}\)

\(=\begin{bmatrix}1\cdot1+1\cdot(-1)+1\cdot(-1)+1\cdot1\\-1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+1\cdot1\\1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+(-1)\cdot1\\3\cdot1+1\cdot(-1)+1\cdot(-1)+(-1)\cdot1\end{bmatrix}= \begin{bmatrix}0\\0\\ 0\\0\end{bmatrix}\)

\(A=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} X=\begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\end{bmatrix}\)

Matrix A , which has the dimensions 4 x 4 , is the matrix of ciefficients of the system , X , which has the dimensions 4 x1 is the matrix of unknowns.

\(AX=0\)

\(\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\end{bmatrix}=\begin{bmatrix}0\\ 0\\0\\ 0\end{bmatrix}\)

If X is a solution of SX=0 , then so is c_1X for any constant c_1 . compute the product AX:

\(AX=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}1\\ -1\\ -1\\1\end{bmatrix}\)

\(=\begin{bmatrix}1\cdot1+1\cdot(-1)+1\cdot(-1)+1\cdot1\\-1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+1\cdot1\\1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+(-1)\cdot1\\3\cdot1+1\cdot(-1)+1\cdot(-1)+(-1)\cdot1\end{bmatrix}= \begin{bmatrix}0\\0\\ 0\\0\end{bmatrix}\)