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. c_1cdotbegin{cases}x_1+x_2+x_3=05x_1-2x_2+2x_3=08x_1+x_2+5x_3=0end{cases}, X=c_1begin{pmatrix}43-7end{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. c_1cdotbegin{cases}x_1+x_2+x_3=05x_1-2x_2+2x_3=08x_1+x_2+5x_3=0end{cases}, X=c_1begin{pmatrix}43-7end{pmatrix}

Question
Matrices
asked 2021-01-16
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 \(c_1\). \(c_1\cdot\begin{cases}x_1+x_2+x_3=0\\5x_1-2x_2+2x_3=0\\8x_1+x_2+5x_3=0\end{cases},\ X=c_1\begin{pmatrix}4\\3\\-7\end{pmatrix}\)

Answers (1)

2021-01-17
Let's write
\(A=\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\ X=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\)
Matrix a, which has the dimensions \(3\times3\), is the matrix of coefficients of the system, X, which has the dimensions \(3\times1\) is the matrix of unknowns.
AX=0
\(\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}\)
If X is a solution of AX=0, then so is \(c_1X\) for any constant c_1. Compute the product AX:
\(AX=\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\begin{bmatrix}4\\3\\-7\end{bmatrix}=\begin{bmatrix}1\cdot4+1\cdot3+1\cdot(-7)\\5\cdot4+(-2)\cdot3+2\cdot(-7)\\8\cdot4+1\cdot3+5\cdot(-7)\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}\)
Result:
\(\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}\)
0

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