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If M is a 3 x 3 real matrix that satisfies M3 = M then
(I) M is invertible
(II) Eigenvalues of M are distinct
(III) M is singular
(I) M is invertible
(II) Eigenvalues of M are distinct
(III) M is singular
Select the correct code
- a)(I) and (II)
- b)(II) and (III)
- c)Only (I)
- d)all are wrong
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If M is a 3 x 3 real matrix that satisfies M3 = M then(I) M is inverti...
To understand why the correct answer is option B, let's first define what it means for a matrix to be invertible, have distinct eigenvalues, and be singular.
- Invertible matrix: A matrix is invertible if there exists another matrix, called its inverse, such that when the two matrices are multiplied together, the result is the identity matrix. In other words, if a matrix M is invertible, then there exists a matrix N such that MN = NM = I, where I is the identity matrix.
- Distinct eigenvalues: The eigenvalues of a matrix are the values λ for which there exists a non-zero vector v such that Mv = λv. If all eigenvalues of a matrix are distinct, it means that no two eigenvalues are the same.
- Singular matrix: A matrix is singular if it is not invertible, which means there is no inverse matrix that satisfies the definition mentioned above.
Now let's analyze the given information that M^3 = M.
1. If M is invertible:
If M is invertible, then we can multiply both sides of the equation M^3 = M by the inverse of M. This gives us M^2 = I, where I is the identity matrix. Multiplying both sides by M again, we get M^3 = IM = M. So, if M is invertible, it satisfies the given equation.
2. If the eigenvalues of M are distinct:
If the eigenvalues of M are distinct, it means that there are no repeated eigenvalues. In this case, we can diagonalize M by finding a diagonal matrix D and an invertible matrix P such that M = PDP^(-1), where D is a diagonal matrix with the eigenvalues of M on the diagonal. Taking the cube of both sides of this equation, we have M^3 = PD^3P^(-1). Since D is a diagonal matrix, we can easily compute D^3 by cubing each diagonal entry. So, if the eigenvalues of M are distinct, it satisfies the given equation.
3. If M is singular:
If M is singular, it means that it is not invertible. In this case, there is no inverse matrix N that satisfies the equation MN = NM = I. Therefore, if M is singular, it does not satisfy the given equation.
Based on the analysis above, we can conclude that both statements (II) and (III) are correct. The correct answer is option B, which states that the eigenvalues of M are distinct and M is singular.
- Invertible matrix: A matrix is invertible if there exists another matrix, called its inverse, such that when the two matrices are multiplied together, the result is the identity matrix. In other words, if a matrix M is invertible, then there exists a matrix N such that MN = NM = I, where I is the identity matrix.
- Distinct eigenvalues: The eigenvalues of a matrix are the values λ for which there exists a non-zero vector v such that Mv = λv. If all eigenvalues of a matrix are distinct, it means that no two eigenvalues are the same.
- Singular matrix: A matrix is singular if it is not invertible, which means there is no inverse matrix that satisfies the definition mentioned above.
Now let's analyze the given information that M^3 = M.
1. If M is invertible:
If M is invertible, then we can multiply both sides of the equation M^3 = M by the inverse of M. This gives us M^2 = I, where I is the identity matrix. Multiplying both sides by M again, we get M^3 = IM = M. So, if M is invertible, it satisfies the given equation.
2. If the eigenvalues of M are distinct:
If the eigenvalues of M are distinct, it means that there are no repeated eigenvalues. In this case, we can diagonalize M by finding a diagonal matrix D and an invertible matrix P such that M = PDP^(-1), where D is a diagonal matrix with the eigenvalues of M on the diagonal. Taking the cube of both sides of this equation, we have M^3 = PD^3P^(-1). Since D is a diagonal matrix, we can easily compute D^3 by cubing each diagonal entry. So, if the eigenvalues of M are distinct, it satisfies the given equation.
3. If M is singular:
If M is singular, it means that it is not invertible. In this case, there is no inverse matrix N that satisfies the equation MN = NM = I. Therefore, if M is singular, it does not satisfy the given equation.
Based on the analysis above, we can conclude that both statements (II) and (III) are correct. The correct answer is option B, which states that the eigenvalues of M are distinct and M is singular.
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If M is a 3 x 3 real matrix that satisfies M3 = M then(I) M is invertible(II) Eigenvalues of M are distinct(III) M is singularSelect the correct codea)(I) and (II)b)(II) and (III)c)Only (I)d)all are wrongCorrect answer is option 'B'. Can you explain this answer?
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