Q 33E Question:聽Consider the matrices... [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 33E (page 98)

Question:Consider the matrices of the form $A = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix})$ , where aand bare arbitrary constants. For which values of aand b is A^-1= A?

Short Answer

Expert verified

$F o r a^{2} + b^{2} = 1, A^{鈭� 1} = A$

Step by step solution

Consider the matrix

Consider the matrix.

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

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,

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

Compute the values of the variables

$A^{鈭� 1} = A$ Compare the matrices (1) and (2).

\begin{array}{l} A^{鈭� 1} = A \\ 鈬� \frac{1}{a^{2} + b^{2}} (\begin{matrix} a & b \\ b & 鈭� a \end{matrix}) = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix}) \\ 鈭� a^{2} + b^{2} = 1 \end{array}

When $a^{2} + b^{2} = 1$ , the condition role="math" localid="1664174540277" $A^{鈭� 1} = A$ gets satisfied.

Hence, for $a^{2} + b^{2} = 1$ , .

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91影视

Question:Consider the matrices of the form $A = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix})$ , where aand bare arbitrary constants. For which values of aand b is A^-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:Consider the matrices of the form A=(abb鈭�a) , where aand bare arbitrary constants. For which values of aand b is A-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

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Logic and Functions

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

Question:Consider the matrices of the form $A = (\begin{matrix} a & b \\ b & 鈭� a \end{matrix})$ , where aand bare arbitrary constants. For which values of aand b is A^-1= A?