Table of Contents
- 1 How do you prove symmetric and skew-symmetric matrix?
- 2 Can a matrix be symmetric and skew-symmetric at the same time?
- 3 Is the matrix A is both symmetric and skew matrix then?
- 4 What is the inverse of a skew-symmetric matrix?
- 5 What are some examples of skew symmetric matrices?
- 6 What happens when identity matrix is added to skew symmetric matrix?
- 7 When a+a is symmetric?
How do you prove symmetric and skew-symmetric matrix?
Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix.
Can a matrix be symmetric and skew-symmetric at the same time?
No they are not one and the same. Skew symmetric matrices are those matrices for which the transpose is the negative of itself but non symmetric matrices do not have this restriction.
What is determinant of a skew symmetric matrix?
Reason : The determinant of a skew symmetric matrix of odd order is equal to zero.
Is the matrix A is both symmetric and skew matrix then?
Thus, the zero matrices are the only matrix, which is both symmetric and skew-symmetric matrix. Hence, option B is correct.
What is the inverse of a skew-symmetric matrix?
Answer is (B) A skew-symmetric matrix if it exists If A is a skew-symmetric matrix of odd order, then |A| = 0. So, inverse does not exist. So, (A-1)T = -A-1 (inverse of a matrix is unique).
How do you know if a matrix is skew symmetric?
A matrix is symmetric if and only if it is equal to its transpose. All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.
What are some examples of skew symmetric matrices?
Some examples of skew symmetric matrices are: When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric. Scalar product of skew-symmetric matrix is also a skew-symmetric matrix. The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.
What happens when identity matrix is added to skew symmetric matrix?
When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible. If A is a skew-symmetric matrix, which is also a square matrix, then the determinant of A should satisfy the below condition:
How do you know if a matrix is a symmetric matrix?
If A is a symmetric matrix, then A = A T and if A is a skew-symmetric matrix then A T = – A. To understand if a matrix is a symmetric matrix, it is very important to know about transpose of a matrix and how to find it.
When a+a is symmetric?
If A is a square matrix then we can write it as the sum of symmetric and skew symmetric matrix. Now given A is a Square matrix,then A+A is also a square matrix. So A+A is symmetric, when A is a square matrix.