Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number c_1 begin{cases}x_1+x_2+x_3+x_4=0-x_1+x_2-x_3+x_4=0 x_1+x_2-x_3-x_4=03x_1+x_2+x_3-x_4=0 end{cases} X =begin{pmatrix}1-1-11end{pmatrix}

Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number c_1 begin{cases}x_1+x_2+x_3+x_4=0-x_1+x_2-x_3+x_4=0 x_1+x_2-x_3-x_4=03x_1+x_2+x_3-x_4=0 end{cases} X =begin{pmatrix}1-1-11end{pmatrix}

Question
Forms of linear equations
asked 2021-03-04
Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number \(c_1\)
\(\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}\)
\(X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}\)

Answers (1)

2021-03-05
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:
\(AX=\begin{bmatrix}1&1&1&1\\-1&1&-1&1\\1&1&-1&-1\\3&1&1&-1\end{bmatrix} \begin{bmatrix}1\\ -1\\ -1\\1\end{bmatrix}\)
\(=\begin{bmatrix}1\cdot1+1\cdot(-1)+1\cdot(-1)+1\cdot1\\-1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+1\cdot1\\1\cdot1+1\cdot(-1)+(-1)\cdot(-1)+(-1)\cdot1\\3\cdot1+1\cdot(-1)+1\cdot(-1)+(-1)\cdot1\end{bmatrix}= \begin{bmatrix}0\\0\\ 0\\0\end{bmatrix}\)
0

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