Question

Let A be ann xx n matrix and let B = A + A^T and C = A − A^T (a) Show that B is symmetric and C is skew symmetric. (b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Matrices
ANSWERED
asked 2021-02-08
Let A be a \(n xx n\) matrix and let
\(B = A + A^T and C = A − A^T\)
(a) Show that B is symmetric and C is skew symmetric.
(b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Expert Answers (1)

2021-02-09
as we know that a matrix E is said to be symmetric if \(E^T=E\) and skew symmetric if \(E^T=−E.\)
as \(B=A+A^T\) therefore,
\(=B^T =(A+A^T)^T\)
\(=A^T+(A^T)^T\)
\(=A^T+A (As (A^T)^T=A )\)
=B
as \(B^T=B\) therefore, B is symmetric matrix.
hence proved.
As C=A−AT
therefore, \(CT=(A−A^T)^T\)
\(=A^T−(A^T)T\)
\(=A^T−A\)
\(=−(A−A^T)\)
=−C
as \(C^T=−C\) therefore, C is skew symmetric matrix. Hence proved.
Now we have to show that every nxxn matrix can be expressed as the sum of the symmetric and skew symmetric matrix. Let A be nxxn matrix. therefore,
\(A=1/2(2A)\)
\(=1/2(A+A)\)
\(=1/2(A+A^T+A−A^T)\)
\(=1/2(B+C)\)
where \(B=A+A^T\) and B is a symmetric matrix and \(C=A−A^T\) and C is a skew symmetric matrix.
herefore,
\(A=1/2(B+C)\)
\(=1/2\)(symmetric matrix +skew symmetric matrix)
therefore, it has been showed that any matrix A of order nxxn can be expressed as the sum of symmetric and skew symmetric matrix.
48
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours
...