Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Let $P_{n}$ be the $n \times n$ matrix whose entries are all ones, except for zeros directly below the main diagonal; for example,
role="math" localid="1659508976827" $P_{5} [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \end{matrix}]$
Find the determinant of $P_{n}$ .

Therefore, the determinant of the given matrix is given by,

$d e t P_{n} = 1, \forall n \in □ .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition.

A determinant is a unique number associated with a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Step 2: Given.

Given matrix,

$P_{5} [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \end{matrix}]$

Step 3: To find determinant.

Using the Laplace expansion along the $n^{t h}$ row, we see that the first $n - 1$ addends are 0, because the corresponding minors for the first 2 elements are 0, since each of them has $n - 2$ columns with all entries 1; and the entry $n^{t h}$ in row and $(n - 1)$ column is 0.

Thus, we're only left with $1 . d e t P_{n - 1}$ . Since $d e t P_{n} = 1, \forall n \in □$ , we have .

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Definition.

Step 2: Given.

Step 3: To find determinant.

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Linear Algebra With Applications

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Short Answer

Let Pn be the n×n matrix whose entries are all ones, except for zeros directly below the main diagonal; for example,role="math" localid="1659508976827" P5[1111101111101111101111101]Find the determinant of Pn .

Step by Step Solution

Step 1: Definition.

Step 2: Given.

Step 3: To find determinant.

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