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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q14E (page 306)

Find the 3-volume of the 3-parallelepiped defined by the vectors
$[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$ .

Short Answer

Expert verified

Therefore,

$\sqrt{d e t [A^{T} A]} = \sqrt{6}$

Step by step solution

01

Matrix Definition.

Matrix is asetof numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an 鈥� m by n 鈥� matrix, written 鈥� $m 脳 n$ 鈥�

02

Given.

Given Matrix,

$[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$

03

Step 3: To find volume of the parallelepiped of the vectors.

The three vectors form the matrix

$A = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 1 & 3 \\ 0 & 1 & 4 \end{matrix}]$ .

Therefore, the area is

$\sqrt{d e t (A^{T} A)} = \sqrt{d e t [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 1 & 3 \\ 0 & 1 & 4 \end{matrix}]} \sqrt{d e t (A^{T} A)} = \sqrt{d e t [\begin{matrix} 1 & 1 & 1 \\ 1 & 4 & 10 \\ 1 & 10 & 30 \end{matrix}]} \sqrt{d e t (A^{T} A)} = \sqrt{6}$

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Most popular questions from this chapter

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

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

6. $[\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 4 & 4 \\ 1 & - 1 & 2 & - 2 \\ 1 & - 1 & 8 & - 8 \end{matrix}]$

In an economics text $^{11}$ we find the following system:

$[\begin{matrix} - R_{1} & R_{1} & - (1 - 伪) \\ 伪 & 1 - 伪 & - {(1 - 伪)}^{2} \\ R_{2} & - R_{2} & \frac{- {(1 - 伪)}^{2}}{伪} \end{matrix}] [\begin{matrix} d x_{1} \\ d y_{1} \\ d p \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ - R_{2} d e_{2} \end{matrix}]$ .

Solve for $d x_{1}, d y_{1}$ , and dp. In your answer, you may refer to the determinant of the coefficient matrix as D. (You need not compute D.) The quantities $R_{1}, R_{2}$ and D are positive, and ais between zero and one. If $d e_{2}$ is positive, what can you say about the signs of $d y_{1}$ and dp?

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

9. $[\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 1 & 3 & 3 & 3 \\ 1 & 1 & 1 & 4 & 4 \\ 1 & 1 & 1 & 1 & 5 \end{matrix}]$

IfA is a $4 脳 4$ matrix whose entries are all 1 or -1 , then $d e t A$ must be divisible by 8 (i.e., $d e t A = 8 k$ for some integer k).

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