Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB - BA is:

Question

Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB - BA is: (A) skewsymmetric (B) symmetric (C) neither symmetric nor skewsymmetric (D) I or - I, where I is an identity matrix

Tardigrade Admin · Accepted Answer

Let A be symmetric matrix and B be skew symmetric matrix.

∴ A^{T} = A

and

B^{T} = - B

Consider

(A B - B A)^{T} = (A B)^{T} - (B A)^{T}

= B^{T} A^{T} - A^{T} B^{T}

= (- B) (A) - (A) (- B)

= - B A + A B = A B - B A

This shows AB - BA is symmetric matrix.

Q. Let A and B be any two 3 × 3 matrices. If A is symmetric and B is skew symmetric, then the matrix AB - BA is:

skewsymmetric

symmetric

neither symmetric nor skewsymmetric

I or - I, where I is an identity matrix

Let A be symmetric matrix and B be skew symmetric matrix. ∴AT=A and BT=−B Consider (AB−BA)T=(AB)T−(BA)T =BTAT−ATBT =(−B)(A)−(A)(−B) =−BA+AB=AB−BA This shows AB - BA is symmetric matrix.

Let A be symmetric matrix and B be skew symmetric matrix.
$∴ A^{T} = A$ and $B^{T} = - B$
Consider
$(A B - B A)^{T} = (A B)^{T} - (B A)^{T}$
$= B^{T} A^{T} - A^{T} B^{T}$
$= (- B) (A) - (A) (- B)$
$= - B A + A B = A B - B A$
This shows AB - BA is symmetric matrix.