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mixer [17]
3 years ago
15

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

Mathematics
1 answer:
Akimi4 [234]3 years ago
6 0

Answer:

The matrix form of given linear system is

\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}

Step-by-step explanation:

Let as assume that

X=\begin{bmatrix}x\\ y\\ z\end{bmatrix}

Given differential equations are

\frac{dx}{dt}=-8x+7y-9z

\frac{dy}{dt}=5x-y

\frac{dz}{dt}=10x+7y+8z

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.

\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}

It can be written as

\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}

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