First slide

Transpose of matrix

Question

If A is symmetric and B is skew-symmetric matrix, then which of the following is/are CORRECT?

Moderate

A

ABA^T is skew-symmetric matrix
B

AB^T + BA^T is symmetric matrix
C

(A + B)(A - B) is skew-symmetric
D

(A + I) (B - I) is symmetric

Solution

$A^{T} = A and B^{T} = - B$
(1) ${({ABA}^{T})}^{T} = {AB}^{T} A^{T} = - ({ABA}^{T})$
(2) ${AB}^{T} + {BA}^{T} = - AB + BA$
$\begin{array}{l} Now, (BA - AB)^{T} = (BA)^{T} - (AB)^{T} \\ = A^{T} B^{T} - B^{T} A^{T} = - AB + BA = {AB}^{T} + {BA}^{T} \end{array}$
(3) $((A + B) (A - B))^{T} = (A - B)^{T} (A + B)^{T}$
$= (A^{T} - B^{T}) (A^{T} + B^{T}) = (A + B) (A - B)$
(4) $((A + I) (B - I))^{T} = (B^{T} - I) (A^{T} + I)$
$= - (- B + I) (A + I) = (B - I) (A + I)$

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