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}

Question
Matrices
asked 2021-03-07
Use the definition of the matrix exponential to compute eA for each of the following matrices:
\(A=\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}\)

Answers (1)

2021-03-08
Step 1 : To determine
Given: \(A=\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}\)
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:
\(A=\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}\)
\(A^2=\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}=\begin{pmatrix}1 & 2 \\0 & 1 \end{pmatrix}\)
\(A^3=\begin{pmatrix}1 & 2 \\0 & 1 \end{pmatrix}\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}=\begin{pmatrix}1 & 3 \\0 & 1 \end{pmatrix}\)
Similarly, \(A^k=\begin{pmatrix}1 & k \\0 & 1 \end{pmatrix}\)
Matrix exponential is given by:
\(e^{At}=\begin{pmatrix}1 & 0 \\0 & 1 \end{pmatrix}+\frac{t}{1!}\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}+\frac{t^2}{2!}\begin{pmatrix}1 & 2 \\0 & 1 \end{pmatrix}+\frac{t^3}{3!}\begin{pmatrix}1 & 3 \\0 & 1 \end{pmatrix}+\dots \frac{t^k}{k!}\begin{pmatrix}1 & k \\0 & 1 \end{pmatrix}+\dots\)
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