Q 6.3-9E ... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 442 pages

ISBN: 9780321796974

Chapter 6: Q 6.3-9E (page 306)

Short Answer

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Definition.

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

To find the relationship between them.

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Most popular questions from this chapter

Use Exercise 31 to find

$d e t [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 9 & 16 & 25 \\ 1 & 8 & 27 & 64 & 125 \\ 1 & 16 & 81 & 256 & 625 \end{matrix}]$

Do not use technology.

Consider two vectors $\overset{⇀}{v}$ and $\overset{⇀}{w}$ in $ℝ^{3}$ . Form the matrix $A = [\overset{⇀}{v} x \overset{⇀}{w} \overset{⇀}{v} \overset{⇀}{w}]$ . Express detA in terms of $|| \overset{⇀}{v} x \overset{⇀}{w} ||$ . For which choices of $\overset{⇀}{v}$ and $\overset{⇀}{w}$ is Ainvertible?

Consider the quadrilateral in the accompanying figure, with vertices $P_{i} = (x_{i}, y_{i})$ , for $i = 1, 2, 3, 4$ . Show that the area of this quadrilateral is

$\frac{1}{2} (d e t [\begin{matrix} x_{1} & x_{2} \\ y_{1} & y_{2} \end{matrix}] + d e t [\begin{matrix} x_{2} & x_{3} \\ y_{2} & y_{3} \end{matrix}] + d e t [\begin{matrix} x_{3} & x_{4} \\ y_{3} & y_{4} \end{matrix}] + d e t [\begin{matrix} x_{4} & x_{1} \\ y_{4} & y_{1} \end{matrix}])$ .

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.

Use Cramer's rule to solve the systems in Exercises 22 through 24.

22. $| \begin{matrix} 3 x + & 7 y = & 1 \\ 4 x + & 11 y = & 3 \end{matrix} |$

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Definition.

To find the relationship between them.

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