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$ A = \left[ {\begin{array}{cc} 1 & 2 & 3 \\ 3 & 7 & 7 \\ 1 & 1 & 3 \end{array} } \right] $

$ B = \left[ {\begin{array}{cc} 1 & 0 & 3 \\ 2 & 1 & 7 \\ 3 & 2 & 9 \end{array} } \right] $

if $ b = \left[ {\begin{array}{cc} b_1 \\ b_2 \\ b_3 \end{array} } \right] $ what condition must be $b_1, b_2, b_3$ satisfy in order that following system to be consistent?

(a): $Ax=b$ (b): $Bx=b$

Anyone can help me to solve this question?

user35603
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user3542501
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2 Answers2

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1) If the question is : Find the $b$ such that there exists $x$ and $y$ such that $Ax=b$ and $By=b$, this is equivalent to $b\in {\rm Im}(A)$ and $b\in {\rm Im}(B)$. As ${\rm det}(A)$ and ${\rm det}(B)$ are not zero, ${\rm Im}(A)={\rm Im}(B)=\mathbb{R}^3$, and there is no condition on $b$.

2) If the question is find the $b$ such that there exists an $x$ such that $Ax=Bx=b$, We see that $(A-B)x=0$. But ${\rm det}(A-B)$ is not zero; hence we must have $x=0$ and $b=0$

Kelenner
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Hint: A short answer is: $b$ should be a linear combination of vectors-columns of matrix $A$ and $b$ should be a linear combination of vectors-columns of matrix $B$.

user35603
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  • How b is linear combination? – user3542501 Aug 04 '14 at 14:16
  • We can rewrite $Ax$ as $\left[ {\begin{array}{cc} 1 & 2 & 3 \ 3 & 7 & 7 \ 1 & 1 & 3 \end{array} } \right] \cdot \left[ {\begin{array}{cc} x_1 \ x_2 \ x_3 \end{array} } \right] = x_1\left[ {\begin{array}{cc} 1 \ 3 \ 1 \end{array} } \right]+x_2\left[ {\begin{array}{cc} 2 \ 7 \ 1 \end{array} } \right]+x_3\left[ {\begin{array}{cc} 3 \ 7 \ 3 \end{array} } \right]$. – user35603 Aug 04 '14 at 14:18