Q 34E Question:聽Consider the diagonal... [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 34E (page 98)

Question:Consider the diagonal matrix
$A = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})$
a. For which values of a, b and c is Ainvertible? If it is invertible, what isA^-1?
b. For which values of the diagonal elements is a diagonal matrix (of arbitrary size) invertible?

Short Answer

Expert verified

a. For the values $a 鈮� 0, b 鈮� 0, c 鈮� 0$ , the matrix is invertible. The inverse matrix is, .

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

b. For $a 鈮� 0, b 鈮� 0, c 鈮� 0$ the diagonal matrix is invertible.

Step by step solution

Consider the matrix.

$A = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}) 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� ...... (1)$

Compute the inverse of the matrix.

The matrix is said to be invertible, if and only if, $r r e f (A) = I^{3}$ .

For the values $a 鈮� 0, b 鈮� 0, c 鈮� 0$ , the matrix is invertible.

Compute the condition.

$\begin{matrix} {(\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})}^{鈭� 1} = [\begin{matrix} a & 0 & 0 & 鈭� 鈥� & 1 & 0 & 0 \\ 0 & b & 0 & 鈭� 鈥� & 0 & 1 & 0 \\ 0 & 0 & c & 鈭� 鈥� & 0 & 0 & 1 \end{matrix}] \\ = [\begin{matrix} 1 & 0 & 0 & 鈭� 鈥� & \frac{1}{a} & 0 & 0 \\ 0 & 1 & 0 & 鈭� 鈥� & 0 & \frac{1}{b} & 0 \\ 0 & 0 & 1 & 鈭� 鈥� & 0 & 0 & \frac{1}{c} \end{matrix}] \end{matrix} T h e r e f o r e, {(\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})}^{鈭� 1} = (\begin{matrix} \frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\ 0 & 0 & \frac{1}{c} \end{matrix})$

Compute the values of the variables

In the case of a diagonal matrix, the non-diagonal elements make the matrix invertible.

That is, $a 鈮� 0, b 鈮� 0, c 鈮� 0$ , and the inverse of the matrix is $A^{鈭� 1} = (\begin{matrix} \frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\ 0 & 0 & \frac{1}{c} \end{matrix})$ .

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91影视

Question:Consider the diagonal matrix
$A = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})$
a. For which values of a, b and c is Ainvertible? If it is invertible, what isA^-1?
b. For which values of the diagonal elements is a diagonal matrix (of arbitrary size) invertible?

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 diagonal matrixA=(a000b000c)a. For which values of a, b and c is Ainvertible? If it is invertible, what isA-1?b. For which values of the diagonal elements is a diagonal matrix (of arbitrary size) invertible?

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

Statistics

Mechanics Maths

Geometry

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

Question:Consider the diagonal matrix
$A = (\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix})$
a. For which values of a, b and c is Ainvertible? If it is invertible, what isA^-1?
b. For which values of the diagonal elements is a diagonal matrix (of arbitrary size) invertible?