Write the given linear system in matrix form. Assume X = x y z . dx dt = −8x + 7y − 9z dy dt = 5x − y dz dt = 10x + 7y + 8z

Answer:The matrix form of given linear system is[tex]\begin{bmatrix}x'\\ y'\\ z'\end{bmatrix}=\begin{bmatrix}-8 & 7 &-9\\ 5 & -1 & 0\\ 10 & 7 & 8\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}[/tex]Step-by-step explanation:Let as assume that[tex]X=\begin{bmatrix}x\\ y\\ z\end{bmatrix}[/tex]Given differential equations are[tex]\frac{dx}{dt}=-8x+7y-9z[/tex][tex]\frac{dy}{dt}=5x-y[/tex][tex]\frac{dz}{dt}=10x+7y+8z[/tex]We need to find the matrix form of given linear system.Write the elements of left side in a column matrix.Write all the coefficients in one matrix first which is called a coefficient matrix.  Multiply coefficient matrix with the variables matrix and equate left and right side.[tex]\begin{bmatrix}\frac{dx}{dt}\\ \frac{dy}{dt}\\ \frac{dz}{dt}\end{bmatrix}=\begin{bmatrix}-8 & 7 &-9\\ 5 & -1 & 0\\ 10 & 7 & 8\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}[/tex]It can be written as[tex]\begin{bmatrix}x'\\ y'\\ z'\end{bmatrix}=\begin{bmatrix}-8 & 7 &-9\\ 5 & -1 & 0\\ 10 & 7 & 8\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}[/tex]