Question #196876

(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


1
Expert's answer
2021-05-25T15:41:06-0400

1)

Given matrix is skew symmetric matrix:

AAA'-A

(00202a3b+c0d3a5b+5c5a8b+6c)=(00d02a3b+c3a5b+5c205a8b+6c)\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=2d=2

2a3b+c=2a+3bc2a-3b+c=-2a+3b-c

    2a3b+c=0\implies 2a-3b+c=0

    3a5b+5c=0\implies 3a-5b+5c=0

    5a+8b6c=0\implies -5a+8b-6c=0

Solving above equations we have:

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


2)

Skew symmetric matrix is

(012103230)\begin{pmatrix} 0 & 1&-2 \\ -1 & 0&3 \\ 2&-3&0 \end{pmatrix}


3)

Let B=A2B=A^2

if A=AT    AAT=A2=BA=A^T\implies AA^T=A^2=B

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

BB is symmetric.

Therefore A2A^2 is symmetric.


4)

det(AT)=ATdet(A^T)=|A^T|

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

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


5)

Given that A matrix is a square matrix,

 let n=3 be the order

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

A=A|-A|=-|A|

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


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS