Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.
$A = [\begin{matrix} 7 & 4 & 2 \\ 5 & 3 & 1 \\ 3 & 1 & 4 \end{matrix}]$

Yes, the given matrix does have an LU factorization.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Matrix Definition.

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are $m$ rows and $n$ columns, the matrix is said to be an “m by n” matrix, written “ $m \times n$ .”

Step 2: Given.

Given matrix,

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

Step 3: To find the given matrix is LU factorization or not.

As seen from Exercise 2.4.93.c, an $n \times n$ matrix has an LU factorization if all of its principal sub matrices are invertible.

We have,

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

To solve,

$d e t (A^{(1)}) = 7 \neq 0 d e t (A^{(2)}) = 1 \neq 0 d e t (A^{(3)}) = 1 \neq 0$

Since, all of its principal sub matrices $A$ are invertible.

Thus, $A$ does have an LU factorization.

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Linear Algebra With Applications

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Short Answer

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.
$A = [\begin{matrix} 7 & 4 & 2 \\ 5 & 3 & 1 \\ 3 & 1 & 4 \end{matrix}]$

Step by Step Solution

Step 1: Matrix Definition.

Step 2: Given.

Step 3: To find the given matrix is LU factorization or not.

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Linear Algebra With Applications

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Short Answer

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.A=[742531314]

Step by Step Solution

Step 1: Matrix Definition.

Step 2: Given.

Step 3: To find the given matrix is LU factorization or not.

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Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.
$A = [\begin{matrix} 7 & 4 & 2 \\ 5 & 3 & 1 \\ 3 & 1 & 4 \end{matrix}]$