Question

Create a situation different from any found in the text that can be represented by a matrix

Short answer

We can determine the kinetic energy of the system using matrices.

Step-by-step solution

Step 1. Create a situation

Let us consider four point masses $m_1,m_2,m_3,m_4$having velocities $v_1,v_2,v_3$and $v_4$respectively.

As we know the kinetic energy of any system is given by the formula $KE=\frac{1}{2}m{v}^2$.

Therefore, to find the kinetic energies of the four masses, we have to convert them into matrix form.

Let the matrix $M$represents the masses and matrix $V$represents the velocities.

$$\begin{gathered} \begin{array}{r}M=\left[\begin{array}{cccc}{m}_1& 0& 0& 0\\ 0& m_2& 0& 0\\ 0& 0& m_3& 0\\ 0& 0& 0& m_4\end{array}\right]\end{array} \\ V=\left[\begin{array}{llll}{v}_1& v_2& v_3& v_4\end{array}\right] \end{gathered}$$

Step 2. Find the Kinetic energy

Therefore,

$K.E=\frac{1}{2}\left[V\right]\left[M\right][V{]}^T$

$$\begin{gathered} K.E=\frac{1}{2}\left[\begin{array}{l}{v}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{v}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{v}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{v}_4\end{array}\right]\left[\begin{array}{cccc}{m}_1& 0& 0& 0\\ 0& m_2& 0& 0\\ 0& 0& m_3& 0\\ 0& 0& 0& m_4\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right] \\ K.E==\frac{1}{2}\left[m_1{v}_1^2+m_2{v}_2^2+m_3{v}_3^2+m_4{v}_4^2\right] \end{gathered}$$

Hence, we have determined the kinetic energy of the system using matrices.