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# Find A,if $\begin{bmatrix}4\\1\\3\end{bmatrix}A=\begin{bmatrix}4 & 8 & 4\\1 & 2 &1\\3 & 6 & 3\end{bmatrix}$

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Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
• If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij}=B_{ij}$, then any element $a_{ij}$ in matrix A is equal to corresponding element $b_{ij}$ in matrix B.
Step1:
Let $B=\begin{bmatrix}4\\1\\3\end{bmatrix}C=\begin{bmatrix}-4 & 8 & 4\\-1 & 2 & 1\\-3 & 6 & 3\end{bmatrix}$
Given:
$\Rightarrow BA=C.$
Since B is a $3\times 1$ matrix.
$\Rightarrow$ A is a $1\times 3$ matrix.
A=[a b c]
Step2:
Now BA=C.
$\begin{bmatrix}4\\1\\3\end{bmatrix}\begin{bmatrix}a & b& c\end{bmatrix}=\begin{bmatrix}-4 & 8 & 4\\-1 & 2 & 1\\-3 & 6 & 3\end{bmatrix}$
$\begin{bmatrix}4a & 4b & 4c\\a & b& c\\3a & 3b & 3c\end{bmatrix}=\begin{bmatrix}-4 & 8 & 4\\-1 & 2 & 1\\-3 & 6 & 3\end{bmatrix}$
Step3:
Since the given two matrices are equal the corresponding elements should be equal.
a=-1,b=2,c=1.
Using the $2^{nd}$ row
$\Rightarrow$ A=[-1 2 1].
answered Mar 21, 2013