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jek_recluse [69]
3 years ago
5

Orthogonally diagonalize the​ matrix, giving an orthogonal matrix P and a diagonal matrix D. To save​ time, the eigenvalues are

-14 and -5.
-9 -4 21 -4 -9 2 2 2 -6

Mathematics
1 answer:
alexgriva [62]3 years ago
3 0

Answer:

P=\left(\begin{array}{ccc}-\frac{2}{3}&-\frac{2}{3}&\frac{1}{3}\\\frac{1}{\sqrt{5}}&0&\frac{2}{\sqrt{5}}\\-\frac{4}{3\sqrt{5}}&\frac{\sqrt{5}}{3}&\frac{2}{3\sqrt{5}}\end{array}\right)

Step-by-step explanation:

It is a result that a matrix A is orthogonally diagonalizable if and only if A is a symmetric matrix.  According with the data you provided the matrix should be

A=\left(\begin{array}{ccc}-9&-4&2\\ -4&-9&2\\2&2&-6\\\end{array}\right)

We know that its eigenvalues are \lambda_{1}=-14, \lambda_{2}=-5, where \lambda_{2}=-5 has multiplicity two.

So if we calculate the corresponding eigenspaces for each eigenvalue we have

E_{\lambda_{1}=-14}=\langle(-2,-2,1)\rangle,E_{\lambda_{2}=-5}=\langle(1,0,2),(-1,1,0)\rangle..

With this in mind we can form the matrices P, D that diagonalizes the matrix A so.

P=\left(\begin{array}{ccc}-2&-2&1\\1&0&2\\-1&1&0\\\end{array}\right)

and

D=\left(\begin{array}{ccc}-14&0&0\\0&-5&0\\0&0&-5\\\end{array}\right)

Observe that the rows of P are the eigenvectors corresponding to the eigen values.

Now you only need to normalize each row of P dividing by its norm, as a row vector.

The matrix you have to obtain is the matrix shown below

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Answer:

5

Step-by-step explanation:

Productivity: My candy bar company made 100 million bars last year, sold for $1 each. I also paid [L] people last year, with an average salary of $100K last year. I have overhead cost of $10M. What was my TOTAL productivity (no units, rounded to 2 decimal places)?

Solution:

Total productivity is the average of labour and capital productivity weighted and adjusted to price fluctuations. It is the ratio of total output to the total input. The total productivity is given by the formula:

Total productivity = total output / total input

Total output = Revenue = number of bars sold * price per bar

Total output = 100 million * $1 = $100 million

Total input = Total salary + overhead cost

Total salary = number of people * average salary = 100 *$100000 = $10 million

overhead cost = $10 million

Total input = $10 million + $10 million = $20 million

Total productivity = total output / total input = $100 million / $20 million

Total productivity = 5

7 0
2 years ago
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Answer:

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Step-by-step explanation:

8 0
2 years ago
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Paladin furnishings generated $4 million in sales during 2021, and its year end total assets were $3.2 million. also, at year en
ElenaW [278]

Without having to raise funds externally sales could increase by $84,507.04.

Given that $4 million sales in 2021 and in end of the year total assets are $3.2, current liabilities are $500,000, payable notes are $200,000, accounts payable is $200,000 and accrued liabilities are $100,000 and in 2022 assets must increase by $.80 for every $1.00 increase in sales.

The amount of sales increase that the company can achieve without having to raise funds externally is calculated by multiplying the current year sales with the self-supporting growth rate.

Current Year Sales = $4,000,000

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Therefore, the Retention Ratio = 50%

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Last Year Total Assets = $3,200,000

Therefore, the Self-supporting Growth Rate=Addition to Retained Earnings÷[Total Assets–Total Spontaneous Liabilities-Addition to Retained Earnings]

= [Last year sales×Profit Margin×(1-Dividend Payout Ratio)]÷[Total Assets-Total Spontaneous Liabilities-Addition to Retained Earnings]

= [$4,000,000×0.03×(1–0.50)]÷[$3,200,000-$300,000–{[$4,000,000×0.04×(1–0.050)}]

= $60,000÷[$3,200,000-$300,000-$60,000]

= $80,000÷$2,840,000

= 0.02112676 or 2.112676%

Therefore, the increase in sales that the company can achieve without having to raise funds externally = Last Year Sales×Self-supporting Growth Rate

= $4,000,000×2.112676%

= $84,507.04

Hence, the Increase in Sales so the company achieve without having to raise funds externally $84,507.04”

Learn about profit margin from here brainly.com/question/5193190

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2 years ago
How to make 5:___=40:32 an equivalent ratio
docker41 [41]

Answer:

4

Step-by-step explanation:

5 : ? = 40 : 32

40 / 5 = 8

32 / 8 = 4

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3 years ago
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