Determine whether the system $A\mathbb x=\mathbb b$ is consistent by examing how $\mathbb b$ relates to the column vectors of $A$.

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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.

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sovon
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sovon

Updated on July 22, 2022

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  • sovon
    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?

    • Abellan
      Abellan over 8 years
      You should show first what you've tried. And then we can help you
  • sovon
    sovon over 8 years
    thanks. i got (a) and (b) . but what about (c) ?? will u kindly make it clear?
  • Ben Grossmann
    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.