umthumaL3e

2022-12-02

With T defined by T(x)=Ax, find a vector x such that T(x)=b
$A=\left(\begin{array}{ccc}1& -3& 2\\ 3& -8& 8\\ 0& 1& 2\\ 1& 0& 8\end{array}\right)\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}b=\left(\begin{array}{c}1\\ 6\\ 3\\ 10\end{array}\right)$
What I have done so far is thatI have merged the two matrices into a single augmented matrix. And row reduced it to get:
$\left(\begin{array}{cccc}1& -3& 2& 1\\ 0& 1& 2& 3\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$
So does this just mean that the answer to the question is $\mathbf{x}=\left(\begin{array}{c}1\\ 3\\ 0\\ 0\end{array}\right)$

Expert

What you now have to do is solve the system of equations
${x}_{1}-3{x}_{2}+2{x}_{3}=1$
${x}_{2}+2{x}_{3}=3$
What happens when you solve for ${x}_{2}$ in the second equation? Hint: (use a parameter, like let ${x}_{3}=t$)

Goundoubuf

Expert

Hint 1: If A has 3 columns, the dimension of x must be $3$
Hint 2: To check your result, compute Ax and see if you got $b$

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