Question 1
1 pts
Calculate $z=(5-2 i)(1+3 i)$.
What is the real part of $z$ ?
Hint: Your answer must be an integer.
Question 3
1 pts
Calculate the eigenvalues of the following matrix:
$$
A=\left(\begin{array}{cc}
4 & 2 \
-3 & -1
\end{array}\right) \text {. }
$$
4 and -1
$2 \pm i$
1 and 2
$-1 \pm 2 i$
Question 5
$1 \mathrm{pts}$
What type of equilibrium is displayed on this phase portrait?
This is a spiral sink.
This is a spiral source.
This is an improper sink.
This is an improper source.
Question 6
1 pts
What is the solution to
$$
\frac{d \mathbf{Y}}{d t}=\left[\begin{array}{ll}
2 & 0 \
0 & 2
\end{array}\right] \mathbf{Y} \text { ? }
$$
$\mathbf{Y}=c_{1} e^{2 t}\left[\begin{array}{c}0 \ -1\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{l}0 \ 1\end{array}\right]$
$\mathbf{Y}=c_{1} e^{-2 t}\left[\begin{array}{l}1 \ 0\end{array}\right]+c_{2} e^{-2 t}\left[\begin{array}{l}0 \ 1\end{array}\right]$
$\mathbf{Y}=\frac{1}{e^{-2 t}}\left[\begin{array}{l}c_{1} \ c_{2}\end{array}\right]$
$\mathbf{Y}=c_{1} e^{-2 t}\left[\begin{array}{l}1 \ 0\end{array}\right]+c_{2} e^{2 t}\left[\begin{array}{l}0 \ 1\end{array}\right]$

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To calculate the product $z = (5 - 2i)(1 + 3i)$, we use the distributive property of multiplication for complex numbers. ‖ ‖ step 2 ⋮ Expand the product: $(5 - 2i)(1 + 3i) = 5 \cdot 1 + 5 \cdot 3i - 2i \cdot 1 - 2i \cdot 3i$. ‖ ‖ step 3 ⋮ Simplify each term: $5 \cdot 1 = 5$, $5 \cdot 3i = 15i$, $-2i \cdot 1 = -2i$, and $-2i \cdot 3i = -6i^2$. ‖ ‖ step 4 ⋮ Recall that $i^2 = -1$, so $-6i^2 = -6(-1) = 6$. ‖ ‖ step 5 ⋮ Combine the real and imaginary parts: $5 + 6 + 15i - 2i = 11 + 13i$. ‖ ‖ step 6 ⋮ The real part of $z$ is $11$. ‖ ∻Question 1 Answer∻ ***A*** ∻Key Concept∻ ⚹Multiplication of Complex Numbers⚹ ∻Explanation∻ ⚹To multiply complex numbers, use the distributive property and combine like terms, remembering that $i^2 = -1$.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To find the eigenvalues of the matrix $A = \left(\begin{array}{cc} 4 & 2 \ -3 & -1 \end{array} ight)$, we solve the characteristic equation $\det(A - \lambda I) = 0$. ‖ ‖ step 2 ⋮ The characteristic equation is $\det\left(\begin{array}{cc} 4 - \lambda & 2 \ -3 & -1 - \lambda \end{array} ight) = 0$. ‖ ‖ step 3 ⋮ Calculate the determinant: $(4 - \lambda)(-1 - \lambda) - (2)(-3) = \lambda^2 - 3\lambda - 2 = 0$. ‖ ‖ step 4 ⋮ Solve the quadratic equation $\lambda^2 - 3\lambda - 2 = 0$ to find the eigenvalues. ‖ ‖ step 5 ⋮ The solutions are $\lambda_1 = 2$ and $\lambda_2 = 1$. ‖ ∻Question 3 Answer∻ ***C*** ∻Key Concept∻ ⚹Eigenvalues of a Matrix⚹ ∻Explanation∻ ⚹Eigenvalues are found by solving the characteristic equation $\det(A - \lambda I) = 0$.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ The given differential equation is $\frac{d \mathbf{Y}}{d t} = \left[\begin{array}{cc} 2 & 0 \ 0 & 2 \end{array} ight] \mathbf{Y}$. ‖ ‖ step 2 ⋮ To solve this, we need to find the eigenvalues and eigenvectors of the matrix $\left[\begin{array}{cc} 2 & 0 \ 0 & 2 \end{array} ight]$. ‖ ‖ step 3 ⋮ The eigenvalues of the matrix are $\lambda_1 = 2$ and $\lambda_2 = 2$. ‖ ‖ step 4 ⋮ The general solution to the differential equation is $\mathbf{Y} = c_1 e^{2t} \left[\begin{array}{c} 1 \ 0 \end{array} ight] + c_2 e^{2t} \left[\begin{array}{c} 0 \ 1 \end{array} ight]$. ‖ ∻Question 6 Answer∻ ***C*** ∻Key Concept∻ ⚹Solving Linear Differential Equations⚹ ∻Explanation∻ ⚹The solution to a system of linear differential equations involves finding the eigenvalues and eigenvectors of the coefficient matrix.⚹◊What are the eigenvalues of the matrix A= $\begin{pmatrix} 4 & 2 \ -3 & -1 \end{pmatrix}$?⍭ Generate me a similar question◊

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