Question

How do you solve an augmented matrix $?$

Answer 1

Solving augmented matrix:

Augmented matrix is derived from the system of linear equations. The coefficients of variables are placed at the first three columns of the augmented matrix and the $4th$column shows the constant terms in the linear system. It is used to simplify a system of linear equations and to find the inverse of a matrix.

For example; Let The three equations of the lines are $a_{1}x+b_{1}y+c_{1}z=d_1,a_{2}x+b_{2}y+c_{2}z=d_2,a_{3}x+b_{3}y+c_{3}z=d_3$.

Let us represent these three equations in the form of an augmented matrix.

$A=\left[\begin{array}{cccc}{a}_1& b_1& \overline{)c_1}& d_1\\ a_2& b_2& \overline{)c_2}& d_2\\ a_3& b_3& \overline{)c_3}& d_3\end{array}\right]$

Now, We apply elementary row operations to make the left side of the bar an identity matrix and the right side gives the solution to the system of equations.

$A=\left[\begin{array}{cccc}1& 0& \overline{)0}& k\\ 0& 1& \overline{)0}& l\\ 0& 0& \overline{)1}& m\end{array}\right]$

Here the elements in the last row represent the values of the variables, and we have $x=k,y=l,z=m,$respectively.

Hence, The augmented matrix is solved through performing rows operations, using the Gauss-Jordan method.