Answer to Question #193226 in Linear Algebra for prince

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 A² 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

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

On comparing we get-

$d=2, \\ 2a-3b+c=-2a+3b-c$

$\Rightarrow 2a-3b+c=0 \\ \Rightarrow 3a-5b+5c=0 \\ \Rightarrow -5a+8b-6c=0$

On solving above equations we have-

a=0,b=0 and c=0

(2) Skew symmetric matrix is-

$A=\begin{bmatrix} 0&1&-2\\-1&0&3\\2&-3&0\end{bmatrix}$

(3) Let B=A^2

   if $A=A^T\Rightarrow A.A^T=A.A=A^2=B$

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

  B is symmetric.

  Therefore $A^2$ is symmetric.

(4)

$det(A^T)=|A^T| \\[9pt] |-A|=(-1)^n|A|=-|A| \\[9pt] det(A^T)=-det(A) \\$

(5) Given that A matrix is a square matrix

 let n=3 be the order

$|-A|=(-1)^3|A|=-|A| \\[9pt] |-A|=-|A|$

det() is an odd function