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If A and B are the matrices of same order such that AB = A and BA = B, then A and B are
- a)Involutory matrices
- b)Idempotent matrices
- c)Orthogonal matrices
- d)Nilpotent matrices
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If A and B are the matrices of same order such that AB = A and BA = B,...
Explanation:
To prove that A and B are idempotent matrices, we need to show that A^2 = A and B^2 = B.
Given: AB = A and BA = B
Proof:
Let's start by proving that A^2 = A.
Multiplying both sides of AB = A by A, we get:
(A)(AB) = (A)A
Now, using the associative property of matrix multiplication, we can write:
(AA)B = AA
Since AB = A, we can substitute AB with A in the above equation:
AA = AA
This implies that A^2 = A, which proves that A is an idempotent matrix.
Similarly, we can prove that B^2 = B.
Multiplying both sides of BA = B by B, we get:
(B)(BA) = (B)B
Using the associative property, we can write:
B(BA) = BB
Substituting BA with B, we have:
B(B) = BB
This implies that B^2 = B, which proves that B is an idempotent matrix.
Therefore, the correct answer is option B: Idempotent matrices.
To prove that A and B are idempotent matrices, we need to show that A^2 = A and B^2 = B.
Given: AB = A and BA = B
Proof:
Let's start by proving that A^2 = A.
Multiplying both sides of AB = A by A, we get:
(A)(AB) = (A)A
Now, using the associative property of matrix multiplication, we can write:
(AA)B = AA
Since AB = A, we can substitute AB with A in the above equation:
AA = AA
This implies that A^2 = A, which proves that A is an idempotent matrix.
Similarly, we can prove that B^2 = B.
Multiplying both sides of BA = B by B, we get:
(B)(BA) = (B)B
Using the associative property, we can write:
B(BA) = BB
Substituting BA with B, we have:
B(B) = BB
This implies that B^2 = B, which proves that B is an idempotent matrix.
Therefore, the correct answer is option B: Idempotent matrices.
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If A and B are the matrices of same order such that AB = A and BA = B, then A and B area)Involutory matricesb)Idempotent matricesc)Orthogonal matricesd)Nilpotent matricesCorrect answer is option 'B'. Can you explain this answer?
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