91Ó°ÊÓ

Step 1: Identity Matrix First, we need the identity matrix of the same order as the given matrix: \(\displaystyle I=\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}\) Step 2: Matrix Powers Compute the powers of the matrix A (remember, we only need a few terms): \(A^2= AA =\left( \begin{array}{rr}1 & 1 \\ -1 & -1 \end{array} \right) \left( \begin{array}{rr}1 & 1 \\ -1 & -1 \end{array} \right)=\left( \begin{array}{rr}0 & 0 \\ 0 & 0 \end{array} \right) \) Since \(A^2\) is the zero matrix, all higher powers will also be the zero matrix, and we can simply stop here. Step 3: Matrix Exponential Compute the matrix exponential using the given definition and the computed matrices: \(e^{A} = I + A + \frac{A^2}{2!}\) Substitute the matrices and simplify: \(e^{A} = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} + \begin{pmatrix} 1 & 1 \\ -1 & -1\end{pmatrix} + \frac{1}{2}\begin{pmatrix}0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ -1 & 0\end{pmatrix}\)

Step 1: Identity Matrix First, we need the identity matrix of the same order as the given matrix: \(\displaystyle I= \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} \) Step 2: Matrix Powers Compute a few matrix powers of the given A: \(A^2= AA=\left( \begin{array}{ll}1 & 1 \\ 0 & 1\end{array} \right) \left( \begin{array}{ll}1 & 1 \\ 0 & 1\end{array} \right) = \begin{pmatrix}1 & 2 \\ 0 & 1\end{pmatrix} \) \(A^3= A^2A=\left( \begin{array}{ll}1 & 2 \\ 0 & 1\end{array} \right) \left( \begin{array}{ll}1 & 1 \\ 0 & 1\end{array} \right) = \begin{pmatrix}1 & 3 \\ 0 & 1\end{pmatrix} \) The powers of A are growing, and we can stop at \(A^3\) since it should give a reasonable estimation. Step 3: Matrix Exponential Compute the matrix exponential using the given definition and the computed matrices: \(e^{A} = I + A + \frac{A^2}{2!} + \frac{A^3}{3!}\) Substitute the matrices and simplify: \(e^{A} = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} + \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix} + \frac{1}{2}\begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix} + \frac{1}{6}\begin{pmatrix}1 & 3 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 + 1 + \frac{1}{2} + \frac{1}{6} & 1 + 1 + \frac{3}{2} + \frac{1}{2} \\ 0 & 1 + 1 + \frac{1}{2} + \frac{1}{6}\end{pmatrix} = \begin{pmatrix}2.6667 & 3 \\ 0 & 2.6667\end{pmatrix} \)

Step 1: Identity Matrix First, we need an identity matrix of the same order as the given matrix: \(\displaystyle I=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\) Step 2: Matrix Powers Compute a few matrix powers of the given A: \(A^2 = AA = \begin{pmatrix}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} = \begin{pmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\) \(A^3 = A^2A = \begin{pmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} = I\) The powers of A are repeating (the exact convergence depends on the matrix), and we can stop at \(A^3\). Step 3: Matrix Exponential Compute the matrix exponential using the given definition and the computed matrices: \(e^{A} = I + A + \frac{A^2}{2!} + \frac{A^3}{3!}\) Substitute the matrices and simplify: \(e^{A} = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} + \begin{pmatrix}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} + \frac{1}{2}\begin{pmatrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} + \frac{1}{6}\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} = \begin{pmatrix}2.5 & 0 & -0.5 \\ 0 & 2.1667 & 0 \\ 0 & 0 & 2.5\end{pmatrix}\)