Question

let A be a skew symmetric matrix of odd order. write the value of ।A।

Answer 1

Answer:

|A| = 0

Step-by-step explanation:

Let, A be a skew-symmetric square matrix of n×n , n odd.

Following general properties of determinants, we have:

$det(A)=det(A^{T})$

Let i and j be the numbers of the rows and columns.

However, since A is a skew-symmetric matrix where

$a_{ij} = - a_{ij}$

Then:

$A^{T}$ = −A

Now, we have that:

det(−A) = det($A^{T}$)

But:

$det(-A) = (-1)^{n} det(A)$ ,  $n = \frac{i}{j}$

∴ det($A^{T}$) = $(-1)^{n}$det(A)

And n is odd, then $(-1)^{n}$ = −1

∴det($A^{T}$) = −det(A)

Then subtracting this equations we have:

∴ 2det(A) = det($A^{T}$) − det($A^{T}$)=0

∴ det(A)=0

Thus, |A| is equal to 0.