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Linear Algebra With Applications
Found in: Page 289
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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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 .

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

detPn=1,n .

See the step by step solution

Step by Step Solution

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,


Step 3: To find determinant.

Using the Laplace expansion along the nth 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 nth in row and n-1 column is 0.

Thus, we're only left with 1.detPn-1 . Since detPn=1,n , we have .

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