Question

# Write the given matrix equation as a system of linear equations without matrices.[(2,0,-1),(0,3,0),(1,1,0)][(x),(y),(z)]=[(6),(9),(5)]

Matrix transformations

Write the given matrix equation as a system of linear equations without matrices.
$$\displaystyle{\left[\begin{matrix}{2}&{0}&-{1}\\{0}&{3}&{0}\\{1}&{1}&{0}\end{matrix}\right]}{\left[\begin{matrix}{x}\\{y}\\{z}\end{matrix}\right]}={\left[\begin{matrix}{6}\\{9}\\{5}\end{matrix}\right]}$$

2021-02-13

Here, $$\displaystyle{\left[\begin{matrix}{2}&{0}&-{1}\\{0}&{3}&{0}\\{1}&{1}&{0}\end{matrix}\right]}{\left[\begin{matrix}{x}\\{y}\\{z}\end{matrix}\right]}={\left[\begin{matrix}{6}\\{9}\\{5}\end{matrix}\right]}$$
the dimension of
$$\displaystyle{\left[\begin{matrix}{2}&{0}&-{1}\\{0}&{3}&{0}\\{1}&{1}&{0}\end{matrix}\right]}$$ is $$3 \cdot 3$$
\$ dimension of $$\displaystyle{\left[\begin{matrix}{x}\\{y}\\{z}\end{matrix}\right]}$$ is $$3 \cdot 1$$
so, the column of $$\displaystyle{\left[\begin{matrix}{2}&{0}&-{1}\\{0}&{3}&{0}\\{1}&{1}&{0}\end{matrix}\right]}$$ matrix is equal to row $$\displaystyle{\left[\begin{matrix}{x}\\{y}\\{z}\end{matrix}\right]}$$
Thus multiplication is possible
by the definition of matrix equality, we have system of linear equations
$$\Rightarrow 2x-z=0$$
$$3y=0$$
$$x+z=0$$