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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?
Most Upvoted Answer
If A and B are the matrices of same order such that AB = A and BA = B,...
Explanation:
Given that AB = A and BA = B, we need to find the type of matrices A and B.
Idempotent Matrices:
An idempotent matrix is a square matrix that, when multiplied by itself, gives the same matrix as the result. That is, AA = A.
Proof:
AB = A
Multiplying both sides with A, we get
ABA = AA
As BA = B, we can replace A in the above equation with BA to get
BAB = AA
Now, we know that AB = A. Substituting this in the above equation, we get
BAA = AA
As AA = A, we can replace AA in the above equation with A to get
BA = A
Now, let's find B^2.
Multiplying both sides of BA = B with B, we get
BAB = B
Substituting AA = A in the above equation, we get
AA(BA) = B
As BA = A, we can replace BA in the above equation with A to get
AA(A) = B
As AA = A, we can replace AA in the above equation with A to get
A^2 = B
Therefore, we have proved that A and B are idempotent matrices.
Hence, the correct option is (B) Idempotent matrices.
Given that AB = A and BA = B, we need to find the type of matrices A and B.
Idempotent Matrices:
An idempotent matrix is a square matrix that, when multiplied by itself, gives the same matrix as the result. That is, AA = A.
Proof:
AB = A
Multiplying both sides with A, we get
ABA = AA
As BA = B, we can replace A in the above equation with BA to get
BAB = AA
Now, we know that AB = A. Substituting this in the above equation, we get
BAA = AA
As AA = A, we can replace AA in the above equation with A to get
BA = A
Now, let's find B^2.
Multiplying both sides of BA = B with B, we get
BAB = B
Substituting AA = A in the above equation, we get
AA(BA) = B
As BA = A, we can replace BA in the above equation with A to get
AA(A) = B
As AA = A, we can replace AA in the above equation with A to get
A^2 = B
Therefore, we have proved that A and B are idempotent matrices.
Hence, the correct option is (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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