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
ANSWERED
asked 2021-02-12

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

Answers (1)

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
\(\Rightarrow 2x-z=0\)
\(3y=0\)
\(x+z=0\)

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