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If A & B are symmetric matrices of same order then AB − BA is

If A & B are symmetric matrices of same order then AB − BA is Correct Answer Skew-symmetric

Concept:

Symmetric Matrix:

  • Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself
  • AT = A or A’ = A


Skew-Symmetric Matrix or Anti-symmetric:

  • Square matrix A is said to be skew-symmetric if aij = −aji for all i and j.
  • Square matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A ⇔ AT = −A
  • Note that all the main diagonal elements in the skew-symmetric matrix are zero.


Transpose of a Matrix:

  • The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.
  • It is denoted by A′or AT


Properties of Transpose Matrix

I. The transpose of the transpose of a matrix is the matrix itself. (AT)T = A

II. The transpose of a matrix times a scalar (k) is equal to the constant times the transpose of the matrix. (kA)T = k AT

III. The transpose of the sum or difference of two matrices is equivalent to the sum or difference of their transposes. (A ± B)T = AT ± BT

IV. The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order.(AB) T = BT AT

V. The determinant of a square matrix is the same as the determinant of its transpose.

Calculation:

Given: A and B are symmetric matrices,

⇒ AT = A and BT = B

Now take (AB – BA) T

We know that the transpose of the sum or difference of two matrices is equivalent to the sum or difference of their transposes.

⇒ (AB – BA) T = (AB) T – (BA) T

The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order.

⇒ (AB – BA) T = (AB) T – (BA) T = BTAT - ATBT

= (BA – AB)

= -(AB – BA)

⇒ (AB – BA) T = -(AB – BA)

∴ AB − BA is Skew-symmetric.

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