Q 60E Question: In Exercises 55 throug... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q 60E (page 100)

Question: In Exercises 55 through 65, show that the given matrix A is invertible, and find the inverse. Interpret the linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ and the inverse transformation $T^{- 1} (\overset{鈫�}{y}) = A^{- 1} \overset{鈫�}{y}$ geometrically. Interpret detA geometrically. In your figure, show the angle胃 and the vectors $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ introduced in Theorem 2.4.10.
60. $[\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .

Short Answer

Expert verified

The inverse of the matrix is, $A^{鈭� 1} = [\begin{matrix} 鈭� 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}] . |\det A| = ||\overset{鈫�}{v}|| 鈥夆赌� |\sin 胃| 鈥夆赌� ||\overset{鈫�}{w}||$ is the area of the parallelogram spanned by $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ .

Step by step solution

Find the determinant

Let be $A = [\begin{matrix} 鈭� 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ the given matrix.

In order to find the inverse of matrix, find the determinant of the matrix.

Therefore,

$\begin{matrix} |A| = |\begin{matrix} 鈭� 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}| \\ = 鈭� 0.64 鈭� 0.36 \\ = 鈭� 1.00 \\ 鈮� 0 \end{matrix}$

$鈭� |A| 鈮� 0$ .

Thus, the inverse of matrix exists.

Find the inverse of matrix

Let us find the inverse of matrix.

$\begin{matrix} A^{鈭� 1} = \frac{1}{|A|} a d j A \\ = \frac{1}{鈭� 1} [\begin{matrix} 0.8 & 鈭� 0.6 \\ 鈭� 0.6 & 鈭� 0.8 \end{matrix}] \\ = [\begin{matrix} 鈭� 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}] \end{matrix}$

To interpret the inverse transformation

Geometrically, $[\begin{matrix} 鈭� 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ is equal to its inverse and it represents a reflection.

Interpret the determinant geometrically

Now, let us interpret the determinant ofA geometrically.

Consider the matrix $A = [\begin{matrix} 鈭� 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .

Let $\overset{鈫�}{v} = [\begin{matrix} 鈭� 0.8 \\ 0.6 \end{matrix}], 鈥夆赌� \overset{鈫�}{w} = [\begin{matrix} 0.6 \\ 0.8 \end{matrix}]$ .

Then $\det A = ||\overset{鈫�}{v}|| \sin 胃 ||\overset{鈫�}{w}||$ , where胃 is the oriented angle from $\overset{鈫�}{v}$ to $\overset{鈫�}{w}$ and $胃 = \frac{蟺}{2}$ .

Thus, $|\det A| = ||\overset{鈫�}{v}|| 鈥夆赌� |\sin 胃| 鈥夆赌� ||\overset{鈫�}{w}||$ is the area of the parallelogram spanned by $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ .

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91影视

Short Answer

Step by step solution

Find the determinant

Find the inverse of matrix

To interpret the inverse transformation

Interpret the determinant geometrically

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