Question
Mathematics Question on Matrices
Let A and B be any two 3 × 3 symmetric and skew symmetric matrices, respectively. Then Which of the following is NOT true?
A
A4 – B4 is a symmetric matrix
B
AB – BA is a symmetric matrix
C
B5 – A5 is a skew-symmetric matrix
D
AB + BA is a skew-symmetric matrix
Answer
B5 – A5 is a skew-symmetric matrix
Explanation
Solution
Given: AT = A, BT = -B
From option 1: Let C = A4 - B4
CT = (A4 - B4) = (A4)T - (B4)T = A4 - B4 = C
From option 2: Let C = AB - BA
CT = (AB - BA)T = (AB)T - (BA)T
= BTAT - ATBT = -BA + AB = C
From option 3: Let C = B5 - A5
CT = (B5 - A5)T = (B5)T - (A5)T = -B5 - A5
From option 4: Let C = AB + BA
CT = (AB + BA)T = (AB)T + (BA)T = -BA - AB = -C
∴ Option 3 is not true.