Question #193226

(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(AT) 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.





1
Expert's answer
2021-05-17T13:41:02-0400

1) Given matrix is skew symmetric matrix-

    A'=-A


 [00202a3b+c0d3a5b+5c5a8b+6c]=[00d02a3b+c3a5b+5c205a8b+6c]\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=2,2a3b+c=2a+3bcd=2, \\ 2a-3b+c=-2a+3b-c


2a3b+c=03a5b+5c=05a+8b6c=0\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=[012103230]A=\begin{bmatrix} 0&1&-2\\-1&0&3\\2&-3&0\end{bmatrix}


(3) Let B=A^2


   if A=ATA.AT=A.A=A2=BA=A^T\Rightarrow A.A^T=A.A=A^2=B

   BT=(ATA)T=(AT)(AT)T=AT.A=A.A=A2=BB^T=(A^TA)^T=(A^T)(A^T)^T=A^T.A=A.A=A^2=B


  B is symmetric.

  Therefore A2A^2 is symmetric.


(4)


det(AT)=ATA=(1)nA=Adet(AT)=det(A)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)3A=AA=A|-A|=(-1)^3|A|=-|A| \\[9pt] |-A|=-|A|


det() is an odd function


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