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Murrr4er [49]
3 years ago
13

) a, p and d are n×n matrices. check the true statements below:

Mathematics
1 answer:
BigorU [14]3 years ago
7 0
A. False. Consider the identity matrix, which is diagonalizable (it's already diagonal) but all its eigenvalues are the same (1).

b. True. Suppose \mathbf P is the matrix of the eigenvectors of \mathbf A, and \mathbf D is the diagonal matrix of the eigenvalues of \mathbf A:


\mathbf P=\begin{bmatrix}\mathbf v_1&\cdots&\mathbf v_n\end{bmatrix}

\mathbf D=\begin{bmatrix}\lambda_1&&\\&\ddots&\\&&\lambda_n\end{bmatrix}

Then

\mathbf{AP}=\begin{bmatrix}\mathbf{Av}_1&\cdots&\mathbf{Av}_n\end{bmatrix}=\begin{bmatrix}\lambda_1\mathbf v_1&\cdots&\lambda_n\mathbf v_n\end{bmatrix}=\mathbf{PD}

In other words, the columns of \mathbf{AP} are \mathbf{Av}_i, which are identically \lambda_i\mathbf v_i, and these are the columns of \mathbf{PD}.

c. False. A counterexample is the matrix

\begin{bmatrix}1&1\\0&1\end{bmatrix}

which is nonsingular, but it has only one eigenvalue.

d. False. Consider the matrix

\begin{bmatrix}0&1\\0&0\end{bmatrix}

with eigenvalue \lambda=0 and eigenvector \begin{bmatrix}k&0\end{bmatrix}^\top, where k\in\mathbb R. But the matrix can't be diagonalized.
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A sector of a circle has a central angle of 100 degrees. If the area of the sector is 50pi, what is the radius of the circle
MrMuchimi

The radius of the circle having the area of the sector 50π, and the central angle of the radius as 100° is <u>6√5 units</u>.

An area of a circle with two radii and an arc is referred to as a sector. The minor sector, which is the smaller section of the circle, and the major sector, which is the bigger component of the circle, are the two sectors that make up a circle.

Area of a Sector of a Circle = (θ/360°) πr², where r is the radius of the circle and θ is the sector angle, in degrees, that the arc at the center subtends.

In the question, we are asked to find the radius of the circle in which a sector has a central angle of 100° and the area of the sector is 50π.

From the given information, the area of the sector = 50π, the central angle, θ = 100°, and the radius r is unknown.

Substituting the known values in the formula Area of a Sector of a Circle = (θ/360°) πr², we get:

50π = (100°/360°) πr²,

or, r² = 50*360°/100° = 180,

or, r = √180 = 6√5.

Thus, the radius of the circle having the area of the sector 50π, and the central angle of the radius as 100° is <u>6√5 units</u>.

Learn more about the area of a sector at

brainly.com/question/22972014

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Step-by-step explanation:

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s^3 or in this case 11^3, its 1331.

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order what?

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