In Problem 17to 20, solve the set of homogeneous equations by row reducing the matrix.

$\{\begin{array}{c}2x-3y+5z=0\\ x+2y-z=0\\ x-5y+6z=0\\ 4x+y+3z=0\end{array}$

A homogeneous system is defined as a set of linear equations in which all of the constants on the right side of the equals sign are equal to zero.

An example of a set of homogenous equations is$\mathbf{\{}\begin{array}{c}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{+}\mathbf{3}\mathbf{z}\mathbf{=}\mathbf{0}\\ \mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{-}\mathbf{6}\mathbf{z}\mathbf{=}\mathbf{0}\\ \mathbf{3}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{y}\mathbf{+}\mathbf{z}\mathbf{=}\mathbf{0}\\ \mathbf{x}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{+}\mathbf{3}\mathbf{z}\mathbf{=}\mathbf{0}\end{array}$ .

Given the set of homogeneous equations$\{\begin{array}{c}2\mathrm{x}-3\mathrm{y}+5\mathrm{z}=0\\ \mathrm{x}+2\mathrm{y}-\mathrm{z}=0\\ \mathrm{x}-5\mathrm{y}+6\mathrm{z}=0\\ 4\mathrm{x}+\mathrm{y}+3\mathrm{z}=0\end{array}$

The matrix of the given set of equations is$\left(\begin{array}{cccc}2& -3& 5& 0\\ 1& 2& -1& 0\\ 1& -3& 6& 0\\ 4& 1& 3& 0\end{array}\right)$.

Do the transformations${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2$and${\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_1$.

$\left(\begin{array}{cccc}1& -5& 6& 0\\ 0& 7& -7& 0\\ 1& -5& 6& 0\\ 4& 1& 3& 0\end{array}\right)$

Do the transformations ${\mathrm{R}}_3\to {\mathrm{R}}_3-{\mathrm{R}}_1$and ${\mathrm{R}}_4\to {\mathrm{R}}_4-4{\mathrm{R}}_1$.

$\left(\begin{array}{cccc}1& -5& 6& 0\\ 0& 7& -7& 0\\ 0& 0& 0& 0\\ 0& 21& -24& 0\end{array}\right)$

Do the transformations ${\mathrm{R}}_4\to \frac{1}{3}{\mathrm{R}}_4$and ${\mathrm{R}}_4\to {\mathrm{R}}_4-{\mathrm{R}}_2$.

$\left(\begin{array}{cccc}1& -5& 6& 0\\ 0& 7& -7& 0\\ 0& 0& 0& 0\\ 0& 0& -1& 0\end{array}\right)$

Do the transformations ${\mathrm{R}}_4\to 6{\mathrm{R}}_4-{\mathrm{R}}_3$and ${\mathrm{R}}_2\to \frac{1}{7}{\mathrm{R}}_2.$

$\left(\begin{array}{cccc}1& -5& 6& 0\\ 0& 1& -1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

Do the transformations ${\mathrm{R}}_1\to {\mathrm{R}}_1+5{\mathrm{R}}_2.$

$\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

Therefore, the set of the given homogenous equations is $x=-z,\mathrm{and}\hspace{0.33em}y=z.$