# Use the definition of the matrix exponential to compute eA for each of the following matrices: A=begin{pmatrix}1 & 1 0 & 1 end{pmatrix}

Use the definition of the matrix exponential to compute eA for each of the following matrices:
$A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$
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Step 1 : To determine
Given: $A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$
To Determine: ${e}^{At}$
Step 2: Calculation
The matrix exponential is defined as:
${e}^{At}=I+\frac{t}{1!}A+\frac{{t}^{2}}{2!}{A}^{2}+\frac{{t}^{3}}{3!}{A}^{3}+\dots$
Calculation:
$A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$
${A}^{2}=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$
${A}^{3}=\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 0& 1\end{array}\right)$
Similarly, ${A}^{k}=\left(\begin{array}{cc}1& k\\ 0& 1\end{array}\right)$
Matrix exponential is given by:
${e}^{At}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\frac{t}{1!}\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)+\frac{{t}^{2}}{2!}\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)+\frac{{t}^{3}}{3!}\left(\begin{array}{cc}1& 3\\ 0& 1\end{array}\right)+\dots \frac{{t}^{k}}{k!}\left(\begin{array}{cc}1& k\\ 0& 1\end{array}\right)+\dots$
Jeffrey Jordon