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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: Q13E (page 306)

Find the area (or 2-volume) of the parallelogram (or 2parallelepiped) defined by the vectors
$[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}] A n d [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$

Short Answer

Expert verified

Therefore, the area of the parallelogram of the vectors is given by,

$\sqrt{d e t (A^{T} A)} = \sqrt{20}$

Step by step solution

01

Definition.

The area of a parallelogram is the number of square units inside the polygon.

The area of a parallelogram can be determined by multiplying the base and height.

For finding the area of a parallelogram, the base and height must be perpendicular. The area of a parallelogram is given by,

Area = b x h

02

Given.

Given Matrix,

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

03

To find area of the parallelogram of the vectors.

These two vectors form the matrix

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

Therefore, the area is

$\sqrt{d e t (A^{T} A)} = \sqrt{d e t ([\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}] [\begin{matrix} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \end{matrix}])} \sqrt{d e t (A^{T} A)} = \sqrt{d e t ([\begin{matrix} 4 & 10 \\ 10 & 30 \end{matrix}])} \sqrt{d e t (A^{T} A)} = \sqrt{20}$

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Most popular questions from this chapter

Consider an nxn matrix A with integer entries such that A=1. Are the entries of $A^{- 1}$ necessarily integers? Explain.

Consider a $4 脳 4$ matrix A with rows ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, {\overset{鈫�}{v}}_{4}$ . If, find the determinants in Exercises 11 through16.

12. localid="1659509477853" $d e t [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \\ {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{4} \end{matrix}]$

In Exercises 62 through 64, consider a function from to that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume that .

64. Using Exercises 62 and 63 as a guide, show that $D (A) = a d - b c = d e t A$ for all $2 脳 2$ matrices A .

There exist real invertible $3 脳 3$ matrices A and S such that .

If two $n 脳 n$ matrices A and B are similar, then the equation $d e t A = d e t B$ must hold.

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