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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.
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let A and B be any two 3 × 3 symmetric and skew-symmetric matric...
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.
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.
Most Upvoted Answer
Let A and B be any two 3 × 3 symmetric and skew-symmetric matric...
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Let A and B be any two 3 × 3 symmetric and skew-symmetric matrices, respectively. Then which of the following isNOTtrue?a)A4- B4is a symmetric matrix.b)AB - BA is a symmetric matrix.c)B5- A5is a skew-symmetric matrix.d)AB + BA is a skew-symmetric matrix.Correct answer is option 'C'. Can you explain this answer?
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