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Snowcat [4.5K]
3 years ago
6

In general, the eigenvalues of a diagonal matrix are given by the entries on the diagonal. Verify this for 2 × 2 and 3 × 3 matri

ces.
Mathematics
1 answer:
egoroff_w [7]3 years ago
7 0

Eigenvalues:

Eigenvalues of a matrix A are all the values of π for the following equation:

d = det (πI-A) = 0

For a 2x2 diagonal matrix:

Let A =\left[\begin{array}{ccc}a1&0\\0&a2\\\end{array}\right] \\and\\I=\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]

\pi I-A=\left[\begin{array}{ccc}\pi-a1&0\\0&\pi-a2\\\end{array}\right] \\

Now,

d = det (πI-A)

π1-a1=0 and π2-a2=0

a1=π1 and a2=π2

Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.

For a 3x3 diagonal matrix:

Let A =\left[\begin{array}{ccc}a1&0&0\\0&a2&0\\0&0&a3\end{array}\right] \\and\\I=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]

\pi I-A=\left[\begin{array}{ccc}\pi-a1&0&0\\0&\pi-a2\\0&0&\pi-a3\end{array}\right] \\

Now,

d = det (πI-A)

π1-a1=0  ;  π2-a2=0  ;  π3-a3=0  

a1=π1   ;   a2=π2   ;   a3=π3

Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.

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umka2103 [35]

Answer:

56.55\ ft^2

Step-by-step explanation:

We know that the diameter of the fountain is 3 feet and that it is in the center of the pond.

We know that the area of a circumference is:

\pi(r)^2

where r is the radius of the circumference.

further

d = 2r

Where d is the diameter of the circumference.

Then the radius r_1 of the source is 1.5 ft.

The radius r_2 of the pond is 1.5 + 3 = 4.5 ft

Finally the area of the pond is equal to the area of the ring that is shown in the image

A = \pi(r_2)^2 - \pi (r_1)^2\\\\A = \pi(4.5)^2 - \pi(1.5)^2\\\\A = \pi(4.5^2 - 1.5^2)\\\\A = 18\pi = 56.55\ ft^2


5 0
3 years ago
Mark has a yard in the shape shown the area of his yard is 100 square meters the bottom side (base) is 20 meters what is the dis
vesna_86 [32]

Answer:

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Step-by-step explanation:

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4 0
2 years ago
Find the volume of a cone with a radius of 8 cm and height 15 cm? Round to the nearest tenth. Please show work.
Nonamiya [84]
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4 0
3 years ago
Find the area of the shaded region
wariber [46]
31x^2 + 53x - 8



Use the FOIL method to find the area of the large rectangle. (7x - 2)(9 + 5x)

First term in each binomial: 7x * 9 = 63x
Outside terms: 7x * 5x = 35x^2
Inside terms: -2 * 9 = -18
Last term in each binomial: -2 * 5x = -10x

Now, just add them all up.

63x + 35x^2 - 8 - 10x

35x^2 + 53x - 8



Now find the area of the square.

2x * 2x = 4x^2



Now, subtract the areas.

31x^2 + 53x - 8



Please consider marking this answer as Brainliest to help me advance.
6 0
3 years ago
What is 150% of 60? <br> Select one:<br><br> 90<br><br> 30<br><br> 9<br><br> 40
Keith_Richards [23]

Answer: 90

Step-by-step explanation:

If you do the following:

100% of 60 = 60

50% of 60 = 30

60 + 30 = 90

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