Consider the matrices -2 -1 1 4 1 A = B = 4 -2 4 -2 -2 4 -2 1 -1 (a) Verify that they are diagonalizable and that they commute.
(b) Find the eigenvalues and eigenvectors of A and verify that its spectrum is nondegenerate. (c) Show that the eigenvectors of A are eigenvectors of B as well.
1 answer:
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Answer:
Part (A) The required volume of the column is .
Part (B) The volume be .
Step-by-step explanation:
Consider the provided information.
It is given that the we have a square with side length "s" lies in a plane perpendicular to a line L.
Also One vertex of the square lies on L.
Part (A)
Suppose there is a square piece of a paper which is attached with a wire through one corner. As you blow it up it spins around on the wire.
This square moves a distance h along L, and generate a corkscrew-like column with square.
The cross section will remain the same.
So the cross section area of original column and the cross section area of twisted column at each point will be the same.
The volume of the column is the area of square times the height.
This can be written as:
Hence, the required volume of the column is .
Part (B) What will the volume be if the square turns twice instead of once?
If the square turns twice instead of once then the volume will remains the same but divide the volume into two equal part.
Hence, the volume be .