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anyanavicka [17]
2 years ago
7

Diagonalize a if possible. (find p and d such that a = pdp−1 for the given matrix

Mathematics
1 answer:
podryga [215]2 years ago
6 0

\mathbf A=\begin{bmatrix}-10&30\\-6&17\end{bmatrix}


Compute the eigenvalues of \mathbf A:


\begin{vmatrix}-10-\lambda&30\\-6&17-\lambda\end{vmatrix}=(-10-\lambda)(17-\lambda)+180=(\lambda-5)(\lambda-2)=0

\implies\lambda=5,\lambda=2


Find the corresponding eigenvectors \eta:


\lambda_1=2\implies\begin{bmatrix}-12&30\\-6&15\end{bmatrix}\eta_1=\mathbf0

\implies\eta_1=\begin{bmatrix}5\\2\end{bmatrix}


\lambda_2=5\implies\begin{bmatrix}-15&30\\-6&12\end{bmatrix}\eta_2=\mathbf0

\implies\eta_2=\begin{bmatrix}2\\1\end{bmatrix}


Now,


\mathbf A=\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}\mathrm{diag}(\lambda_1,\lambda_2)\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}^{-1}

\mathbf A=\begin{bmatrix}5&2\\2&1\end{bmatrix}\begin{bmatrix}2&0\\0&5\end{bmatrix}\begin{bmatrix}5&2\\2&1\end{bmatrix}^{-1}

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Answer:

The percent of the people who tested positive actually have the disease is 38.64%.

Step-by-step explanation:

Denote the events as follows:

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<em>N</em> = the test result is negative

Given:

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Compute the value of P (P|X) as follows:

P(P|X)=1-P(P|X^{c})=1-0.15=0.85

Compute the probability of a positive test result as follows:

P(P)=P(P|X)P(X)+P(P|X^{c})P(X^{c})\\=(0.85\times0.10)+(0.15\times0.90)\\=0.22

Compute the probability of a person having the disease given that he/she was tested positive as follows:

P(X|P)=\frac{P(P|X)P(X)}{P(P)}=\frac{0.85\times0.10}{0.22} =0.3864

The percentage of people having the disease given that he/she was tested positive is, 0.3864 × 100 = 38.64%.

3 0
2 years ago
Compute: (3.5· 2/7 − 2/3 )÷16
velikii [3]

Answer:

≅ 0.02083333

Step-by-step explanation:

(3.5 * 2/7 - 2/3)÷16 = 1

48

≅ 0.02083333

1: Conversion a decimal number to a fraction: 3.5

2: Multiple: 3.5 * 2

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= 7 · 2

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= 1 · 14

1 · 14

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3: Subtract:

4: Divide

<em><u>Hope this helps.</u></em>

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Each swing that Susan builds requires $10 in hardware and $20 in wood. She uses the expression 10 + 20 to represent the total co
jenyasd209 [6]
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m_a_m_a [10]
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Step-by-step explanation:

4:25*100 =  

(4*100):25 =  

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3 years ago
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