Determine whether the system $A\mathbb x=\mathbb b$ is consistent by examing how $\mathbb b$ relates to the column vectors of $A$.
For example: in question a), the equation $Ax = b$ becomes $$ 2x_1 + x_2 = 3\\ -2x_1 - x_2 = 1 $$ Note, however, that the second row of $A$ is a multiple of the first row of $A$. In particular, $$ (-1)[2x_1 + x_2] = (-1)[3] \implies\\ -2x_1 - x_2 = -3 $$ So, our system equations tells us that $-2x_1 - x_2 = 1$ and that $-2x_1 - x_2 = -3$. However, it is impossible for $-2x_1 - x_2$ to equal $1$ and $-3$. We therefore conclude that the system of equations is inconsistent.
The other questions can be answered similarly.
sovon
Updated on July 22, 2022Comments
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sovon over 1 year
10. For each of the choices of $A$ and $\mathbb b$ that follow, determine whether the system $A\mathbb x=\mathbb b$ is consistent by examing how $\mathbb b$ relates to the column vectors of $A$. Explain your answers in each case.
(a) $$A=\begin{bmatrix}2&1\\-2&-1\end{bmatrix}\qquad\mathbb b=\begin{bmatrix}3\\1\end{bmatrix}$$
(b) $$A=\begin{bmatrix}1&4\\2&3\end{bmatrix}\qquad\mathbb b=\begin{bmatrix}5\\5\end{bmatrix}$$
(c) $$A=\begin{bmatrix}3&2&1\\3&2&1\\3&2&1\end{bmatrix}\qquad\mathbb b=\begin{bmatrix}1\\0\\-1\end{bmatrix}$$
I do not understand this. How can I solve these?
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Abellan over 8 yearsYou should show first what you've tried. And then we can help you
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sovon over 8 yearsthanks. i got (a) and (b) . but what about (c) ?? will u kindly make it clear?
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Ben Grossmann over 8 years@sovon (c) is even easier; there's no multiplying to do. Set up the system of equations and you should immediately find a contradiction.