Answer to Question #196876 in Linear Algebra for mpopo

Question

(6.1) Find the values of a, b and c such the matrix below is skew symmetric.

0 0 d

0 2a − 3b + c 3a − 5b + 5c

2 0 5a − 8b + 6c

(6.2) Give an example of a skew symmetric matrix.

(6.3) Prove that A (4) 2 is symmetric whenever A is skewsymmetric.

(6.4) Determine an expression for det(A) in terms of det(A (T ) if A is a square skewsymmetric.

(6.5) Assume that A has an odd number of rows and also an odd number of columns. In this particular case, show that det(·) is an odd function

Answer 1

1)

Given matrix is skew symmetric matrix:

$A'-A$

$\begin{pmatrix} 0 & 0 &2\\ 0 & 2a-3b+c &0\\ d & 3a-5b+5c &5a-8b+6c\\ \end{pmatrix}=-\begin{pmatrix} 0 & 0 &d\\ 0 & 2a-3b+c &3a-5b+5c\\ 2 & 0 &5a-8b+6c\\ \end{pmatrix}$

On comparing we get:

$d=2$

$2a-3b+c=-2a+3b-c$

$\implies 2a-3b+c=0$

$\implies 3a-5b+5c=0$

$\implies -5a+8b-6c=0$

Solving above equations we have:

$a=0,b=0,c=0$

2)

Skew symmetric matrix is

$\begin{pmatrix} 0 & 1&-2 \\ -1 & 0&3 \\ 2&-3&0 \end{pmatrix}$

3)

Let $B=A^2$

if $A=A^T\implies AA^T=A^2=B$

$B^T=(A^TA)^T=A^T(A^T)^T=A^TA=A^2=B$

$B$ is symmetric.

Therefore $A^2$ is symmetric.

4)

$det(A^T)=|A^T|$

$|-A|=(-1)^n|A|=-|A|$

$det(A^T)=-det(A)$

5)

Given that A matrix is a square matrix,

 let n=3 be the order

$|-A|=(-1)^3|A|=-|A|$

$|-A|=-|A|$

So, $det()$ is an odd function.