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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 442 pages

ISBN: 9780321796974

Chapter 6: Q 6.3-5E (page 306)

Question: The tetrahedron defined by three vectorsin R³is the set of all vectors of the form, whereand . Explain why the volume of this tetrahedron is one sixth of the volume of the parallelepiped defined by.

Short Answer

Expert verified

Therefore,

, which is one sixth of the volume of the given parallelepiped.

Step by step solution

Known.

To explain the volume of tetrahedron is one sixth of the volume of parallelepiped.

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Most popular questions from this chapter

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

4. $[\begin{matrix} 1 & - 1 & 2 & - 2 \\ - 1 & 2 & 1 & 6 \\ 2 & 1 & 14 & 10 \\ - 2 & 6 & 10 & 33 \end{matrix}]$

$d e t (4 A) = 4 d e t A$ for all $4 \times 4$ matricesA.

In an economics text, $^{10}$ we find the following system:

$s Y + a r = I ° + G m Y - h r = M_{s} - M$

Solve for Y and r.

If all the entries of an invertible matrixA are integers, then the entries of $A^{- 1}$ must be integers as well.

There exists a $4 \times 4$ matrix whose entries are all 1or -1 , and such that $d e t A = 16$ .

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Question: The tetrahedron defined by three vectorsin R³is the set of all vectors of the form, whereand . Explain why the volume of this tetrahedron is one sixth of the volume of the parallelepiped defined by.

Short Answer

Step by step solution

Known.

To explain the volume of tetrahedron is one sixth of the volume of parallelepiped.

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Most popular questions from this chapter

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Question: The tetrahedron defined by three vectorsin R3is the set of all vectors of the form, whereand . Explain why the volume of this tetrahedron is one sixth of the volume of the parallelepiped defined by.

Short Answer

Step by step solution

Known.

To explain the volume of tetrahedron is one sixth of the volume of parallelepiped.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Applied Mathematics

Calculus

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Question: The tetrahedron defined by three vectorsin R³is the set of all vectors of the form, whereand . Explain why the volume of this tetrahedron is one sixth of the volume of the parallelepiped defined by.