Find the determinant of the n × n matrix. |1 - n11111 1- n11111 .1 -1

The determinant of the n × n matrix. 1-n11…111-n1…1⋮⋮⋮…⋮111…1-n

Answered: |1 - n 1 1 1 1 1 1- n 1 1 1 1 1 . 1 -1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the determinant of the n × n matrix.

Expert Solution

Step 1 To find:

The determinant of the n × n matrix.

$[\begin{matrix} 1 - n & 1 & 1 & \dots & 1 \\ 1 & 1 - n & 1 & \dots & 1 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ \\ 1 & 1 & 1 & \dots & 1 - n \end{matrix}]$

Step 2

Applying $R_{1} \to R_{1} + R_{2} + R_{3}$

$[\begin{matrix} n (1) - n & n (1) - n & n (1) - n & \dots & n (1) - n \\ 1 & 1 - n & 1 & \dots & 1 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ \\ 1 & 1 & 1 & \dots & 1 - n \end{matrix}] (There are 1 n's in each column) = [\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 1 & 1 - n & 1 & \dots & 1 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ \\ 1 & 1 & 1 & \dots & 1 - n \end{matrix}]$

It is known that if an entire row or an entire column of a matrix consists of zeros, then its determinant is zero.

Hence, the determinant of the given matrix is zero.

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