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If A and B are symmetric matrices of the same order, then which one of the following is not correct?
- a)A + B is a symmetric matrix
- b)B — BA is a symmetric matrix.
- c)AB + BA is a symmetric matrix
- d)A + AT and B + BT are symmetric matrices.
Correct answer is option 'B'. Can you explain this answer?
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If A and B are symmetric matrices of the same order, then which one of...
Understanding Symmetric Matrices
Symmetric matrices are matrices that are equal to their transpose. If A and B are symmetric matrices, meaning A = AT and B = BT, we can analyze the options given in the question to identify which statement is incorrect.
Option A: A + B is a symmetric matrix
- A + B = (A + B)T
- Using properties of transposes: (A + B)T = AT + BT = A + B
- Thus, this statement is correct.
Option B: B – BA is a symmetric matrix
- We need to check if (B - BA)T = B - BA.
- (B - BA)T = BT - (BA)T = BT - ATB = B - ATB (since B and A are symmetric)
- This does not necessarily equal B - BA, as ATB is not equal to BA in general.
- Thus, this statement is incorrect.
Option C: AB + BA is a symmetric matrix
- We check: (AB + BA)T = BTAT + ABT = AB + BA (since A and B are symmetric).
- Thus, this statement is correct.
Option D: A + AT and B + BT are symmetric matrices
- A + AT = A + A = 2A (which is symmetric), and similarly for B.
- Thus, this statement is correct.
Conclusion
The incorrect statement is option B, as B - BA is not guaranteed to be symmetric due to the non-commutative nature of matrix multiplication.
Symmetric matrices are matrices that are equal to their transpose. If A and B are symmetric matrices, meaning A = AT and B = BT, we can analyze the options given in the question to identify which statement is incorrect.
Option A: A + B is a symmetric matrix
- A + B = (A + B)T
- Using properties of transposes: (A + B)T = AT + BT = A + B
- Thus, this statement is correct.
Option B: B – BA is a symmetric matrix
- We need to check if (B - BA)T = B - BA.
- (B - BA)T = BT - (BA)T = BT - ATB = B - ATB (since B and A are symmetric)
- This does not necessarily equal B - BA, as ATB is not equal to BA in general.
- Thus, this statement is incorrect.
Option C: AB + BA is a symmetric matrix
- We check: (AB + BA)T = BTAT + ABT = AB + BA (since A and B are symmetric).
- Thus, this statement is correct.
Option D: A + AT and B + BT are symmetric matrices
- A + AT = A + A = 2A (which is symmetric), and similarly for B.
- Thus, this statement is correct.
Conclusion
The incorrect statement is option B, as B - BA is not guaranteed to be symmetric due to the non-commutative nature of matrix multiplication.
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If A and B are symmetric matrices of the same order, then which one of the following is not correct?a)A + B is a symmetric matrixb)B — BA is a symmetric matrix.c)AB + BA is a symmetric matrixd)A + AT and B + BT are symmetric matrices.Correct answer is option 'B'. Can you explain this answer?
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