Aufopferaq

2021-11-27

A(t) is a $2×2$ matrix of differentiable functions and X(t) is a $2×1$ column matrix of differentiable functions, prove the product rule

Ched1950

Step1
Let The matrices A(t) and X(t) be
$A\left(\begin{array}{c}t\end{array}\right)=\left[\begin{array}{cc}{A}_{11}\left(t\right)& {A}_{12}\left(t\right)\\ {A}_{21}\left(t\right)& {A}_{22}\left(t\right)\end{array}\right]$
$X\left(\begin{array}{c}t\end{array}\right)=\left[\begin{array}{cc}{x}_{1}\left(t\right)& \\ {x}_{2}\left(t\right)\end{array}\right]$
The product rule of differentiation states that
$\frac{d}{dt}\left[A\left(t\right)X\left(t\right)\right]={A}^{\prime }\left(t\right)X\left(t\right)+A\left(t\right){X}^{\prime }\left(t\right)$
Step2
$A\left(\begin{array}{c}t\end{array}\right)X\left(\begin{array}{c}t\end{array}\right)=\left[\begin{array}{cc}{A}_{11}\left(t\right)& {A}_{12}\left(t\right)\\ {A}_{21}\left(t\right)& {A}_{22}\left(t\right)\end{array}\right]\left[\begin{array}{cc}{x}_{1}\left(t\right)& \\ {x}_{2}\left(t\right)\end{array}\right]$
$=\left[\begin{array}{cc}{A}_{11}\left(t\right){x}_{1}\left(t\right)& {A}_{12}\left(t\right){x}_{2}\left(t\right)\\ {A}_{21}\left(t\right){x}_{1}\left(t\right)& {A}_{22}\left(t\right){x}_{2}\left(t\right)\end{array}\right]$
Now,
$\frac{d}{dt}\left[A\left(t\right)X\left(t\right)\right]$
$=\left[\begin{array}{cc}\frac{d}{dt}{A}_{11}\left(t\right){x}_{1}\left(t\right)& {A}_{12}\left(t\right){x}_{2}\left(t\right)\\ \frac{d}{dt}{A}_{21}\left(t\right){x}_{1}\left(t\right)& {A}_{22}\left(t\right){x}_{2}\left(t\right)\end{array}\right]$
$=\left[\begin{array}{cc}\frac{d}{dt}\left[{A}_{11}\left(t\right){x}_{1}\left(t\right)& {A}_{12}\left(t\right){x}_{2}\left(t\right)\right]\\ \frac{d}{dt}\left[{A}_{21}\left(t\right){x}_{1}\left(t\right)& {A}_{22}\left(t\right){x}_{2}\left(t\right)\right]\end{array}\right]$
$=\left[\begin{array}{cc}{A}_{11}^{\prime }\left(t\right){x}_{1}\left(t\right)+{A}_{11}\left(t\right){x}_{1}^{\prime }\left(t\right)+& {A}_{12}^{\prime }\left(t\right){x}_{2}\left(t\right)+{A}_{12}\left(t\right){x}_{2}^{\prime }\left(t\right)\\ {A}_{21}^{\prime }\left(t\right){x}_{1}\left(t\right)+{A}_{21}\left(t\right){x}_{1}^{\prime }\left(t\right)+& {A}_{22}^{\prime }\left(t\right){x}_{2}\left(t\right)+{A}_{22}\left(t\right){x}_{2}^{\prime }\left(t\right)\end{array}\right]$
Step 3
Now, ${A}^{\prime }\left(\begin{array}{c}t\end{array}\right)=\left[\begin{array}{cc}{A}_{11}^{\prime }\left(t\right)& {A}_{12}^{\prime }\left(t\right)\\ {A}_{21}\left(t\right)& {A}_{22}\left(t\right)\end{array}\right]$

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