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Expert-verified Found in: Page 289 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Let ${{\mathbit{P}}}_{{\mathbf{n}}}$ be the ${\mathbit{n}}{\mathbf{×}}{\mathbit{n}}$ matrix whose entries are all ones, except for zeros directly below the main diagonal; for example,role="math" localid="1659508976827" ${{\mathbit{P}}}_{{\mathbf{5}}}\mathbf{\left[}\begin{array}{ccccc}\mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}$Find the determinant of ${{\mathbit{P}}}_{{\mathbf{n}}}$ .

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

$det{P}_{n}=1,\forall n\in \square .$

See the 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,

${{P}}_{{5}}\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 0& 1& 1& 1& 1\\ 1& 0& 1& 1& 1\\ 1& 1& 0& 1& 1\\ 1& 1& 1& 0& 1\end{array}\right]$

## Step 3: To find determinant.

Using the Laplace expansion along the ${n}^{th}$ 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 ${n}^{th}$ in row and $\left(n-1\right)$ column is 0.

Thus, we're only left with $1.det{P}_{n-1}$ . Since $det{P}_{n}=1,\forall n\in \square$ , we have . ### Want to see more solutions like these? 