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

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:
$$e^{At}=\begin{pmatrix}1 & 0 \\0 & 1 \end{pmatrix}+\frac{t}{1!}\begin{pmatrix}1 & 1 \\0 & 1 \end{pmatrix}+\frac{t^2}{2!}\begin{pmatrix}1 & 2 \\0 & 1 \end{pmatrix}+\frac{t^3}{3!}\begin{pmatrix}1 & 3 \\0 & 1 \end{pmatrix}+\dots \frac{t^k}{k!}\begin{pmatrix}1 & k \\0 & 1 \end{pmatrix}+\dots$$

### Relevant Questions

Use the definition of the matrix exponential to compute eA for each of the following matrices:
$$A=\begin{bmatrix}1 & 0&-1 \\0 & 1&0\\0&0&1 \end{bmatrix}$$
Write out the system of equations that corresponds to each of the following augmented matrices:
(a)$$\begin{pmatrix}3 & 2&|&8 \\1 & 5&|&7 \end{pmatrix}$$
(b)$$\begin{pmatrix}5 & -2&1&|&3 \\2 & 3&-4&|&0 \end{pmatrix}$$
(c)$$\begin{pmatrix}2 & 1&4&|&-1 \\4 & -2&3&|&4 \\5 & 2&6&|&-1 \end{pmatrix}$$
(d)$$\begin{pmatrix}4 & -3&1&2&|&4 \\3 & 1&-5&6&|&5 \\1 & 1&2&4&|&8\\5 & 1&3&-2&|&7 \end{pmatrix}$$
Compute the product AB by the definition of the product of​ matrices, where $$Ab_1 \text{ and } Ab_2$$ are computed​ separately, and by the​ row-column rule for computing AB.
$$A=\begin{bmatrix}-1 & 2 \\2 & 5\\5&-3 \end{bmatrix} , B=\begin{bmatrix}4 & -1 \\-2 & 4 \end{bmatrix}$$
Determine the product AB
AB=?
For each of the pairs of matrices that follow, determine whether it is possible to multiply the first matrix times the second. If it is possible, perform the multiplication.
$$\begin{bmatrix}1 & 4&3 \\0 & 1&4\\0&0&2 \end{bmatrix}\begin{bmatrix}3 & 2 \\1 & 1\\4&5 \end{bmatrix}$$
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$$\begin{pmatrix}1 & 2 &-3&-1 \\ -2 & -4 & 6 &3 \end{pmatrix}$$
Determine the null space of each of the following matrices:
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Let $$A=\begin{bmatrix}1 & 2 \\-1 & 1 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}$$
a)Find elementary matrices $$E_1 \text{ and } E_2$$ such that $$C=E_2E_1A$$
b)Show that is no elementary matrix E such that C=EA
Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
$$A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}$$
If the operation is not possible , write NOT POSSIBLE and be able to explain why
a)A+B
b)B+C
c)2A
Given the matrix
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}$$
and suppose that we have the following row reduction to its PREF B
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}$$
Write $$A \text{ and } A^{-1}$$ as product of elementary matrices.
$$A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}$$
(i) $$AX^T=BC^3$$
(ii) $$A^{-1}(X-T)^T=(B^{-1})^T$$
(iii) $$XF=F^{-1}-D^T$$