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If A and B are square matrices of the same order, then(A+B)2 = A2+2AB+B2 implies
- a)AB + BA = O
- b)AB = O
- c)AB = BA
- d)none of these.
Correct answer is 'A'. Can you explain this answer?
Verified Answer
If A and B are square matrices of the same order, then(A+B)2= A2+2AB+B...
If A and B are square matrices of same order , then , product of the matrices is not commutative.Therefore , the given result is true only when AB = BA.
Most Upvoted Answer
If A and B are square matrices of the same order, then(A+B)2= A2+2AB+B...
Community Answer
If A and B are square matrices of the same order, then(A+B)2= A2+2AB+B...
Explanation:
To prove that the correct answer is option (a), we need to show that the given equation (A B)^2 = A^2 + 2AB + B^2 implies AB + BA = O, where O represents the zero matrix.
Expanding the given equation:
(A B)^2 = A^2 + 2AB + B^2
Expanding the left-hand side of the equation:
(A B)^2 = (A B)(A B)
Using the distributive property of matrix multiplication, we can expand this as:
(A B)(A B) = A(A B) + B(A B)
Now, using the associative property of matrix multiplication, we can further expand this as:
A(A B) + B(A B) = A^2 + AB + BA + B^2
Simplifying the equation:
Now, equating the expanded equation with the given equation:
A^2 + AB + BA + B^2 = A^2 + 2AB + B^2
Subtracting A^2 and B^2 from both sides:
AB + BA = AB
Subtracting AB from both sides:
BA = O
Thus, we have proved that AB + BA = O, which is the required result to show that the correct answer is option (a).
Conclusion:
Therefore, the correct answer is option (a): AB + BA = O.
To prove that the correct answer is option (a), we need to show that the given equation (A B)^2 = A^2 + 2AB + B^2 implies AB + BA = O, where O represents the zero matrix.
Expanding the given equation:
(A B)^2 = A^2 + 2AB + B^2
Expanding the left-hand side of the equation:
(A B)^2 = (A B)(A B)
Using the distributive property of matrix multiplication, we can expand this as:
(A B)(A B) = A(A B) + B(A B)
Now, using the associative property of matrix multiplication, we can further expand this as:
A(A B) + B(A B) = A^2 + AB + BA + B^2
Simplifying the equation:
Now, equating the expanded equation with the given equation:
A^2 + AB + BA + B^2 = A^2 + 2AB + B^2
Subtracting A^2 and B^2 from both sides:
AB + BA = AB
Subtracting AB from both sides:
BA = O
Thus, we have proved that AB + BA = O, which is the required result to show that the correct answer is option (a).
Conclusion:
Therefore, the correct answer is option (a): AB + BA = O.
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