Q 32E Question:聽Find all matrices聽 (... [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 32E (page 98)

Question:Find all matrices $(\begin{matrix} a & b \\ c & d \end{matrix})$ such that $a d 鈭� b c = 1$ and $A^{鈭� 1} = A$ .

Short Answer

Expert verified

The matrices which satisfied $a d 鈭� b c = 1$ and $A^{鈭� 1} = A$ are, $(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), (\begin{matrix} 鈭� 1 & 0 \\ 0 & 鈭� 1 \end{matrix})$ .

Step by step solution

Consider the matrix

Consider the matrix.

$A = (\begin{matrix} a & b \\ c & d \end{matrix}) 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� ...... (1)$

The matrix is said to be invertible if and only if the determinant of the matrix is nonzero.

The inverse of the matrix is,

${(\begin{matrix} a & b \\ c & d \end{matrix})}^{鈭� 1} = \frac{1}{a d 鈭� b c} (\begin{matrix} d & 鈭� b \\ 鈭� c & a \end{matrix})$

where.

$d e t (\begin{matrix} a & b \\ c & d \end{matrix}) = a d 鈭� b c$

Compute the inverse of the matrix

The inverse of the matrix is,

$\begin{matrix} A^{鈭� 1} = {(\begin{matrix} a & b \\ c & d \end{matrix})}^{鈭� 1} \\ = \frac{1}{a d 鈭� b c} (\begin{matrix} d & 鈭� b \\ 鈭� c & a \end{matrix}) \\ = (\begin{matrix} d & 鈭� b \\ 鈭� c & a \end{matrix}) 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€夆€夆€夆€夆€夆€夆€夆€夆€� 鈭� a d 鈭� b c = 1 \\ 鈭� A^{鈭� 1} = (\begin{matrix} d & 鈭� b \\ 鈭� c & a \end{matrix}) 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� ...... (2) \end{matrix}$

Compute the values of the variables

Compare the matrices (1) and (2).

$\begin{matrix} A^{鈭� 1} = A \\ (\begin{matrix} d & 鈭� b \\ 鈭� c & a \end{matrix}) = (\begin{matrix} a & b \\ c & d \end{matrix}) \\ 鈭� a = d, b = c = 0 \end{matrix}$

Compute the matrices using the above values.

$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), (\begin{matrix} 鈭� 1 & 0 \\ 0 & 鈭� 1 \end{matrix})$

Hence, the matrices which satisfy ad - bc = 1 and A^-1= A are, $(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), (\begin{matrix} 鈭� 1 & 0 \\ 0 & 鈭� 1 \end{matrix})$ .

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91影视

Question:Find all matrices $(\begin{matrix} a & b \\ c & d \end{matrix})$ such that $a d 鈭� b c = 1$ and $A^{鈭� 1} = A$ .

Short Answer

Step by step solution

Consider the matrix

Compute the inverse of the matrix

Compute the values of the variables

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91影视

Question:Find all matrices (abcd) such that ad鈭�bc=1 and A鈭�1=A .

Short Answer

Step by step solution

Consider the matrix

Compute the inverse of the matrix

Compute the values of the variables

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Decision Maths

Applied Mathematics

Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Question:Find all matrices $(\begin{matrix} a & b \\ c & d \end{matrix})$ such that $a d 鈭� b c = 1$ and $A^{鈭� 1} = A$ .