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Chapter 6: Q15E (page 306)

Demonstrate Theorem 6.3.6 for linearly dependent vectorv鈬赌1,....,v鈬赌m.

Short Answer

Expert verified

Therefore, the area of the parallelepiped is given by,

detATA=0.

Step by step solution

01

Definition. 

Generally, the parallelepiped is a three-dimensional geometric solid with six faces that are parallelograms.

Parallelepiped is a three-dimensional solid shape.

It has 6 faces, 12 edges, and 8 vertices.

All faces of a parallelepiped are in the shape of a parallelogram.

02

To demonstrate the linear dependent vectors. 

Ifv鈬赌1,v鈬赌2,.....,v鈬赌m,鈭�鈩�2are linearly dependent, then the matrix A that they form has linearly dependent columns.

Therefore,ATAis non-invertible.

So, the area of the parallelepiped isgiven by

detATA=0.

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Most popular questions from this chapter

Even if an n脳n matrix A fails to be invertible, we can define the adjoint adj(A)as in Theorem 6.3.9. The ijthentry of adj(A)is (-1)i+jdet(Aji). For which n脳nmatrices A isadj(A)=0 ? Give your answer in terms of the rank ofA. See Exercise 41.

In an economics text,10we find the following system:

sY+ar=I掳+GmY-hr=Ms-M

Solve for Y and r.

Consider a 4x4 matrix A with rows V1鈫�,V2鈫�,V3鈫�,V4鈫�. If det(A) = 8, find the determinants in Exercises 11 through 16.

16. role="math" localid="1659506283449" det[6v鈬赌1+2v鈬赌4v鈬赌2v鈬赌33v鈬赌1+v鈬赌4]

If all the diagonal entries of an n脳nmatrixA are even integers and all the other entries are odd integers, then Amust be an invertible matrix.

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.

A=[742531314]
See all solutions

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