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]](https://tex.z-dn.net/?f=Let%20A%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Da1%260%5C%5C0%26a2%5C%5C%5Cend%7Barray%7D%5Cright%5D%20%5C%5Cand%5C%5CI%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%5C%5C%5Cend%7Barray%7D%5Cright%5D)
![\pi I-A=\left[\begin{array}{ccc}\pi-a1&0\\0&\pi-a2\\\end{array}\right] \\](https://tex.z-dn.net/?f=%5Cpi%20I-A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Cpi-a1%260%5C%5C0%26%5Cpi-a2%5C%5C%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
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]](https://tex.z-dn.net/?f=Let%20A%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Da1%260%260%5C%5C0%26a2%260%5C%5C0%260%26a3%5Cend%7Barray%7D%5Cright%5D%20%5C%5Cand%5C%5CI%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%260%5C%5C0%261%260%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%5D)
![\pi I-A=\left[\begin{array}{ccc}\pi-a1&0&0\\0&\pi-a2\\0&0&\pi-a3\end{array}\right] \\](https://tex.z-dn.net/?f=%5Cpi%20I-A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Cpi-a1%260%260%5C%5C0%26%5Cpi-a2%5C%5C0%260%26%5Cpi-a3%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
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.
Answer:
5.641 = x
Step-by-step explanation:
Work on the right triangle
pythag theorem
9^2 + L^2 = 10^2
L = sqrt 19
then x = 10 - sqrt 19 =5.641
If philani buys a scooter for 15,000 and wants to pay 15% down then he must pay 2,250. After this they will still need to pay 12,750 to pay of the scooter, but after the 10% annually (for two years) then they will owe 15,750 for the scooter rather than 12,750. Hope this helps!
Answer:
1) ∠A=84°
2) ∠C=20°
Step-by-step explanation:
1)
First, find ∠C:
<em>(I'm assuming the exterior angle of 126° makes a straight line with ∠C)</em>
The angles on a straight line always add up to 180. Therefore:
∠C+126=180
∠C=180-126
∠C=54
Then find ∠B:
We also know that all the angles in a triangle add up to 180. Therefore:
∠A+∠B+∠C=180
∠A+∠B+54=180
∠A+∠B=126
<em>(we know ∠A=2(∠B))</em>
2(∠B)+∠B=126
3(∠B)=126
∠B=42
Now, find ∠A:
∠A=2(∠B)
∠A=2(42)
∠A=84°
2)
First, find ∠B:
<em>(Again, I'm assuming the exterior angle of 100° makes a straight line with ∠B)</em>
The angles on a straight line always add up to 180. Therefore:
∠B+100=180
∠B=180-100
∠B=80
Then find ∠A:
We also know that all the angles in a triangle add up to 180. Therefore:
∠A+∠B+∠C=180
∠A+80+∠C=180
∠A+∠C=100
<em>(we know ∠A=4(∠C))</em>
4(∠C)+∠C=100
5(∠C)=100
∠C=20°
Here, Twice fast means, she takes half time in doing same amount of work.
Let, Anna takes = 2t & Jane takes = t
It would be: 1/t + 1/2t = 1/15
2t + t / 2t² = 1/15
15(3t) = 2t²
2t² = 45t
Divide both sides by t,
2t = 45
t = 45/2
t = 22.5
& 2t = 22.5(2) = 45
In short, Anna would take 45 minutes
Hope this helps!
