Question

Construct a 5 × 5 real matrix A such that its elements aij are given by the relation

aij = 0 ∀ i = j

aij = 1 ∀ i 6= j

(a) Calculate the determinant of the matrix A by special techniques taught in the class.

(b) Is matrix A is a singular or non-singular ?​

Answer 1

Given : aij = 0  i = j

aij =1  i ≠ j

To Find : Construct a 5 × 5 matrix

determinant of the matrix A

matrix A is a singular or non-singular  

Solution:

aij = 0  i = j

Hence all diagonal elements

a₁₁ , a₂₂ , a₃₃ , a₄₄ , a₅₅ = 0

All non diagonal elements will be 1.

Hence 5 × 5  Matrix is :

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

Determinant = 4

As Determinant  is non zero

Hence matrix A  is non-singular  

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