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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: Q56E (page 292)

Use the characterization of the determinant given in Exercise 55 to show that $d e t (A M) = (d e t A) (d e t M)$ .

Short Answer

Expert verified

Therefore,

D (A) = \det A, 鈭赌 A 鈭� {鈩�}^{n 脳 n}

Step by step solution

01

Definition.

A determinant is a unique number associated with asquare 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.

02

To show D(A)=det(A)

For a fixed invertible matrix $M 鈭� {鈩�}^{n 脳 n}$ , let $D (A) = \frac{\det (A M)}{\det M}$ .

We have to prove the properties $a ., b .$ and $c$ of Ex. 55 to see that, in fact

$D (A) = \det A$ .

We have,

$D (A) = \frac{\det (A M)}{\det M}$

$D (A) = \frac{1}{\det M} \det (A M)$

a. Since the determinant is linear in the columns, then $D$ is also linear in the columns.

b. Since the determinant changes sign when two columns are switched, then $D$ does the same.

c. We compute

$D (I_{n}) = \frac{\det (I_{n} M)}{\det M} D (I_{n}) = \frac{\det M}{\det M} D (I_{n}) = 1 .$

Therefore,

$D (A) = \det A, 鈭赌 A 鈭� {鈩�}^{n 脳 n}$ .

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Most popular questions from this chapter

If $\overset{鈫�}{u}, \overset{鈫�}{v}, \overset{鈫�}{w}$ are three unit vectors in ${鈩�}^{3}$ , what are the possible values of $d e t [\begin{matrix} \overset{鈫�}{u} & \overset{鈫�}{v} & \overset{鈫�}{w} \end{matrix}]$ ?

For an invertiblenxnmatrix A, what is the relationship betweenadj(A)and $a d j (A^{- 1})$ a?

Question:We say that a linear transformation Tfrom $N^{3} to N^{3}$ to preserves orientation if it transforms any positively oriented basis into another positively oriented basis. See Exercise 19. Explain why a linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ preserves orientation if (and only if) detAis positive.

29. If $A = [\overset{鈫�}{u} \overset{鈫�}{v} \overset{鈫�}{w}]$ is a $3 脳 3$ matrix, then the formula $d e t (A) = \overset{鈫�}{V} . (\overset{鈫�}{u} 脳 \overset{鈫�}{w})$ must hold.

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.

$A = [\begin{matrix} 7 & 4 & 2 \\ 5 & 3 & 1 \\ 3 & 1 & 4 \end{matrix}]$

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