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anyanavicka [17]
3 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]3 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:

Option B.

Step-by-step explanation:

The given exponential equation is

y=ab^x  ...(1)

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The graph passes through the point (-3,24), so substitute  x=-3 and y=24 in equation (1).

24=ab^{-3}   ...(2)

The graph passes through the point (-2,12), so substitute  x=-2 and y=12 in equation (1).

12=ab^{-2}   ...(3)

Divide equation (3) by equation (2).

\dfrac{12}{24}=\dfrac{ab^{-2}}{ab^{-3}}

0.5=b

Substitute b=0.5 in equation (2).

24=a(0.5)^{-3}

24=\dfrac{a}{(0.5)^{3}}

24=\dfrac{a}{0.125}

24\times 0.125=a

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Substitute a=3 and b=0.5 in equation (1).

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8 0
3 years ago
Caroline's hair is
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The inequality can be used to determine how many months will it take for Caroline's hair to be at least as long as Trinity's hair is (1/5)x + 7 ≥ (1/10)x + 10

<h3>What is an equation? </h3>

An equation is an expression that shows the relationship between two or more numbers and variables. An independent variable is a variable that does not depend on other variables while a dependent variable is a variable that depends on other variables.

Let x represent the number of months.

For Caroline's hair:

Length = (1/5)x + 7

For Trinity's hair:

Length = (1/10)x + 10

For Caroline's hair to be at least as long as Trinity's hair, hence:

(1/5)x + 7 ≥ (1/10)x + 10

The inequality can be used to determine how many months will it take for Caroline's hair to be at least as long as Trinity's hair is (1/5)x + 7 ≥ (1/10)x + 10

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2 years ago
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3 years ago
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The expressions that are equivalent when m = 1 and m = 4 is;

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In both cases, the expressions are equivalent and as such option B is the right one.

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4 0
3 years ago
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