eAugmentedMatrix

Example Find the augmented matrix of the following system of equations.．

$\displaystyle \left\{\begin{array}{cccccc} a_{11}x_{1} &+& a_{12}x_{2}&+\cdots+... ... a_{m1}x_{1} &+& a_{m2}x_{2}&+\cdots+&a_{mn}x_{n}& = b_{m} \end{array}\right.$

Answer The matrix composed of the coefficients $A = (a_{ij})$ is called Coefficient Matrix．The matrix composed of the coefficients and the constant terms denoted by $[A : {\bf b}]$ is called Augmented Matrix. If we denote $[A : {\bf b}]$ augmented matrix, ${\mathbf x} = (x_{j})$ $n$ th dimensional column vector, ${\bf b} = (b_{i})$ , $m$ th dimensional column vector, then the above system of linear equations can be expressed by

$\displaystyle A{\mathbf x} = {\bf b}\ \ \ \mbox{or}\ \ \ [A : {\bf b}]$

Thus, the augmented matrix is as follows:

$\displaystyle [A:{\bf b}] = \begin{pmatrix} a_{11} & a_{12} &\cdots& a_{1n} & b... ...cdots& \vdots & \vdots\\ a_{m1} & a_{m2} &\cdots& a_{mn} & b_{m} \end{pmatrix}$

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