The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. displaystyle{left[begin{matrix}{1}&{0}&-{1}&-{2}{0}&{1}&{2}&{3}end{matrix}right]}

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The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. displaystyle{left[begin{matrix}{1}&{0}&-{1}&-{2}{0}&{1}&{2}&{3}end{matrix}right]}

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The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
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The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$

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A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio $x_1$
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asked 2021-03-04

Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations.
$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$

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Answer 1

The given matrix is
$\displaystyle{B}={\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$
Reduce the given augmented matrix B in to system of linear equation
Ax=b
The matrix form of the first equation is
$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}\\{0}&{1}&{2}\end{matrix}\right]}\cdot{\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\end{matrix}\right]}={\left[\begin{matrix}-{2}\\{3}\end{matrix}\right]}$
where
$\displaystyle{A}={\left[\begin{matrix}{1}&{0}&-{1}\\{0}&{1}&{2}\end{matrix}\right]},{x}={\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\end{matrix}\right]},{b}={\left[\begin{matrix}-{2}\\{3}\end{matrix}\right]}$
We use another equation to find a. general solution
$x_1-x_3=-2 (3)$
$x_2+2x_3=3 (4)$
$x_3=x_3 (5)$
Then
$\displaystyle{\left({3}\right)}\Rightarrow{x}_{{1}}=-{2}+{x}_{{3}}$
$\displaystyle{\left({4}\right)}\Rightarrow{x}_{{2}}={3}-{2}{x}_{{3}}$
$\displaystyle{\left({5}\right)}\Rightarrow{x}_{{3}}={x}_{{3}}$
In vectors form, the general solution, we obtain
$\displaystyle{x}={\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\end{matrix}\right]}={\left[\begin{matrix}-{2}+{x}_{{3}}\\{3}-{2}{x}_{{3}}\\{x}_{{3}}\end{matrix}\right]}={\left[\begin{matrix}-{2}\\{3}\\{0}\end{matrix}\right]}+{x}_{{3}}{\left[\begin{matrix}{1}\\-{2}\\{1}\end{matrix}\right]}$
answer $\displaystyle{x}={\left[\begin{matrix}-{2}\\{3}\\{0}\end{matrix}\right]}+{x}_{{3}}{\left[\begin{matrix}{1}\\-{2}\\{1}\end{matrix}\right]}$

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The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. displaystyle{left[begin{matrix}{1}&{0}&-{1}&-{2}{0}&{1}&{2}&{3}end{matrix}right]}

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. displaystyle{left[begin{matrix}{1}&{0}&-{1}&-{2}{0}&{1}&{2}&{3}end{matrix}right]}

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