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Let A be an n × n real matrix such that A2 = I and y = be an n – dimensional vector. Then the linear system of equations Ax = y has
- a)No sol utio n
- b)a unique solution
- c)More than one but finitely many independent solutions
- d)Infinitely many independent solutions
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let A be an n × n real matrix such that A2 = I and y = be an n &#...
By Cramer’s rule AX =y has unique solution.
Most Upvoted Answer
Let A be an n × n real matrix such that A2 = I and y = be an n &#...
Explanation:
To determine the number of solutions to the linear system of equations Ax = y, we need to analyze the properties of matrix A.
Given that A^2 = I, where A is an n × n real matrix, we can conclude that A is an involutory matrix. An involutory matrix is a matrix that, when multiplied by itself, yields the identity matrix.
Properties of an involutory matrix:
1. Eigenvalues: The eigenvalues of an involutory matrix are either 1 or -1.
2. Eigenvectors: The eigenvectors corresponding to eigenvalue 1 form the null space (kernel) of A - I, and the eigenvectors corresponding to eigenvalue -1 form the null space of A + I.
3. Rank: The rank of an involutory matrix is equal to its trace, which is the sum of its eigenvalues.
Determining the number of solutions:
Since A is an involutory matrix and not equal to the identity matrix, it must have eigenvalues other than 1. Therefore, the linear system of equations Ax = y will have more than one solution.
However, since A is not a singular matrix (A ≠ 0), the null space of A is only the zero vector. Therefore, the linear system of equations Ax = y will have a unique solution.
Answer:
The correct answer is option B) a unique solution.
To determine the number of solutions to the linear system of equations Ax = y, we need to analyze the properties of matrix A.
Given that A^2 = I, where A is an n × n real matrix, we can conclude that A is an involutory matrix. An involutory matrix is a matrix that, when multiplied by itself, yields the identity matrix.
Properties of an involutory matrix:
1. Eigenvalues: The eigenvalues of an involutory matrix are either 1 or -1.
2. Eigenvectors: The eigenvectors corresponding to eigenvalue 1 form the null space (kernel) of A - I, and the eigenvectors corresponding to eigenvalue -1 form the null space of A + I.
3. Rank: The rank of an involutory matrix is equal to its trace, which is the sum of its eigenvalues.
Determining the number of solutions:
Since A is an involutory matrix and not equal to the identity matrix, it must have eigenvalues other than 1. Therefore, the linear system of equations Ax = y will have more than one solution.
However, since A is not a singular matrix (A ≠ 0), the null space of A is only the zero vector. Therefore, the linear system of equations Ax = y will have a unique solution.
Answer:
The correct answer is option B) a unique solution.
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Let A be an n × n real matrix such that A2 = I and y = be an n – dimensional vector. Then the linear system of equations Ax = y hasa)No sol utio n b)a unique solution c)More than one but finitely many independent solutions d)Infinitely many independent solutionsCorrect answer is option 'B'. Can you explain this answer?
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