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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: Q2E (page 305)

Find the area of the triangle defined by $[\begin{matrix} 3 \\ 7 \end{matrix}]$ and $[\begin{matrix} 8 \\ 2 \end{matrix}]$ .

Short Answer

Expert verified

Therefore, the Area of triangle is 25 .

Step by step solution

01

Definition.

Basically, it is equal to half of the Area of the parallelogram.

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle.

$A = 1 / 2 脳 b 脳 h$

02

Given.

Given Matrix,

$[\begin{matrix} 3 \\ 7 \end{matrix}]$

And

$[\begin{matrix} 8 \\ 2 \end{matrix}]$

03

To find the area of the triangle.

Since the area of a triangle formed by two vectors is equal to half the area of a parallelogram formed by the same vectors, we compute

$\frac{1}{2} |\det [\begin{matrix} 3 & 8 \\ 7 & 2 \end{matrix}]| = \frac{1}{2} . 50 |dett [\begin{matrix} 3 & 8 \\ 7 & 2 \end{matrix}]| = 25$

Therefore,

Area of triangle is 25 .

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Most popular questions from this chapter

Find all 2x2matrices for which $[\begin{matrix} 2 \\ 3 \end{matrix}]$ is an eigenvector with associated eigenvalue -1.

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role="math" localid="1659502796300" $a . F (A) = b c b . F (A) = c d c . F (A) = a c d . F (A) = b c - a d e . F (A) = c$

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Question: Let A and B be 2 x 2 matrices with integer entries such that A,A+B,A+2B,A+3B, and A+4B are all invertible matrices whose inverses have integer entries. Show that A+5B is invertible and that it鈥檚 inverse has integer entries. This question was in the William Lowell Putnam Mathematical Competition in 1994. Hint: Consider the function F(t)=(det(A+tB))² -1. Shows that this is a polynomial; what can you say about its degree? Find the values f(0), f(1), f(2), f(3), f(4),using Exercise 53. Now you can determine f(t) by using a familiar result: If a polynomial f(t) of degree <m has more than m zeros, then f(t)=0 for all t.

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